# CRM CAMP in Nonlinear Analysis

The main goal of the CRM CAMP project is to bring together the worldwide community of researchers in the area of computer-assisted methods of proof, especially those working in the areas of dynamical systems theory and nonlinear analysis. This community has enjoyed dramatic growth over the last three decades, and has developed methods to resolve a number of important unsolved problems in mathematics. Yet participating researchers are scattered around the globe, and there is a growing need for a regular forum for discussion and dissemination of results. This is especially important in current time of unprecedented travel interruption.

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# Seminar Series

Weekly Seminar Series: every Tuesdays of the summer at 10am (Montreal/Miami time).

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# Past seminars Change display

June 29, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Computer-assisted proofs for renormalisation fixed-points and eigenfunctions for period-doubling universality in maps of the interval

We prove the existence of a fixed point to the renormalisation operator for period doubling in maps of even degree at the critical point. We work with a modified operator that encodes the action of the renormalisation operator on even functions. Building on previous work, our proof uses rigorous computer-assisted means to bound operations in a space of analytic functions and hence to show that a quasi-Newton operator for the fixed-point problem is a contraction map on a suitable ball.

We bound the spectrum of the Frechet derivative of the renormalisation operator at the fixed point, establishing the hyperbolic structure, in which the presence of a single essential expanding eigenvalue explains the universal asymptotically self-similar bifurcation structure observed in the iterations of families of maps lying in the relevant universality class.

By recasting the eigenproblem for the Frechet derivative in a particular nonlinear form, we again use the contraction mapping principle to gain rigorous bounds on eigenfunctions and their corresponding eigenvalues. In particular, we gain tight bounds on the eigenfunction corresponding to the essential expanding eigenvalue delta. We adapt the procedure to the eigenproblem for the scaling of added uncorrelated noise.

Our computations use multi-precision interval arithmetic with rigorous directed rounding modes to bound tightly the coefficients of the relevant power series and their high-order terms, and the corresponding universal constants.

Computer-assisted proofs for renormalisation fixed-points and eigenfunctions for period-doubling universality in maps of the interval

June 22, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Combinatorial Topological Dynamics

Since the publication in 1998 of the seminal work by Robin Forman on combinatorial Morse theory there has been growing interest in dynamical systems on finite spaces. The main motivation to study combinatorial dynamics comes from data science. But, they also provide very concise models of dynamical phenomena and show some potential in certain computer assisted proofs in dynamics.

In the talk I will present the basic ideas of Conley theory for combinatorial dynamical system, particularly for a combinatorial multivector field which is a generalization of combinatorial vector field introduced by Forman. The theory is based on concepts which are analogous to the concepts of classical theory: isolating neighborhood, isolated invariant set, index pair, Conley index, Morse decomposition, connection matrix. The concepts are analogous but in some cases surprisingly different in details. This may be explained by the non-Hausdorff nature of combinatorial topological spaces.

Despite the differences there seem to be strong formal ties between the combinatorial and classical dynamics on topological level. A Morse decomposition of a combinatorial vector field on an abstract simplicial complex induces a semiflow on the geometric realization of the complex with a Morse decomposition exhibiting the same Conley-Morse graph. Actually, this correspondence of Morse decompositions and Conley-Morse graphs applies to every semiflow which is transversal to the boundaries of top dimensional cells of a certain cellular decomposition of the phase space associated with the combinatorial vector field.

There is also a formal relation in the opposite direction. Given a smooth flow and a cellular decomposition of its phase space which is transversal to the flow, there is an induced combinatorial multivector field on the cellular structure of the phase space. Moreover, if the induced combinatorial multivector field admits a periodic trajectory with an appropriate Conley index, a periodic orbit exists also for the original smooth flow.

The formal ties seem to provide a natural framework for a rigorous global analysis of the dynamics of a flow: the decomposition into the gradient and recurrent part together with the computation of the Conley-Morse graph, connection matrix and revealing the internal structure of the recurrent part.

Based on joint work with J. Barmak, T. Dey, M. Juda, T. Kaczynski, T. Kapela, J. Kubica,  M. Lipiński  R. Slechta, R. Srzednicki, J. Thorpe and Th. Wanner.

Combinatorial Topological Dynamics

June 15, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Chaos in the Chua's circuit

Several results on the existence of chaos in the Chua's circuit with piesewise linear and cubic nonlinearities will be presented.

Chaos in the Chua's circuit

June 8, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### OPEN PROBLEMS SERIES: Some conjectures that seem difficult to prove

In many cases, numerics and/or heuristics provide compelling evidence for statements that we have trouble proving rigorously. I will discuss some examples, mostly inspired by fluid flows.

OPEN PROBLEMS SERIES: Some conjectures that seem difficult to prove

June 1, 2021 from 10:00 to 10:00 (Montreal/EST time) Zoom meeting

### CRM CAMP Colloquium: Encounters with Computer-Assisted Proofs in Early Days

The talk recounts how computer-assisted proofs came into two theorems on the quantum mechanics of Coulomb systems during the 1980's.

CRM CAMP Colloquium: Encounters with Computer-Assisted Proofs in Early Days

May 18, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Bifurcation Points in the Ohta-Kawasaki Model

Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki, which is a Cahn-Hilliard-like equation together with a nonlocal term. Unlike the Cahn-Hilliard model, even on one-dimensional spatial domains the steady state bifurcation diagram of the Ohta-Kawasaki model is still not fully understood. We therefore present computer-assisted proof techniques which can be used to validate and continue its bifurcation points. This includes not only fold points, but also pitchfork bifurcations which are the result of a cyclic group action beyond forcing through Z_2 symmetries.

Bifurcation Points in the Ohta-Kawasaki Model

May 11, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### A computer assisted proof of Wright's conjecture: counting and discounting slowly oscillating periodic solutions to a DDE

A classical example of a nonlinear delay differential equation is Wright's equation: y'(t)=\alpha y(t−1)[1 + y(t)], considering \alpha>0 and y(t)>-1. This talk discusses two conjectures associated with this equation: Wright's conjecture, which states that the origin is the global attractor for all \alpha \in (0,\pi/2]; and Jones' conjecture, which states that there is a unique slowly oscillating periodic solution for \alpha>\pi/2.

To prove Wright's conjecture our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at \alpha=\pi/2. Using a rigorous numerical integrator we characterize slowly oscillating periodic solutions and calculate their stability, proving Jones' conjecture for \alpha \in [1.9,6.0] and thereby all \alpha \ge 1.9. We complete the proof of Jones conjecture using global optimization methods, extended to treat infinite dimensional problems.

A computer assisted proof of Wright's conjecture: counting and discounting slowly oscillating periodic solutions to a DDE

May 4, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Rigorous numerics of blow-up solutions for autonomous ODEs

In this talk, I willl discuss about recent studies concerning rigorous numerics of blow-up solutions for autonomous ODEs in a systematic way under a mild assumption of vector fields. The fundamental tools used here are “compactifications” of phase spaces which map the infinity to the boundary of transformed phase spaces (the “horizon”), and “time-scale desingularizations determined by the original vector fields”. Blow-up solutions are then essentially transformed into solutions on stable manifolds of invariant sets on the horizon. In particular, rigorous enclosures of blow-up solutions and their blow-up times can be validated by means of standard machineries of dynamical systems such as ODE integrators, locally defined Lyapunov functions and parameterization of invariant manifolds. Dynamical system approach shown here reveals many quantitative and qualitative nature of blow-up behavior for various concrete dynamical systems. A series of works presented in the present talk (involving rigorous numerics) are based on joint works with Profs. Akitoshi Takayasu, Nobito Yamamoto and Jean-Philippe Lessard.

Rigorous numerics of blow-up solutions for autonomous ODEs

April 27, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### A computer assisted counterexample to Payne’s nodal line conjecture with few holes

Payne conjectured in 1967 that the nodal line of the second Dirichlet eigenfunction must touch the boundary of the domain. In their 1997 breakthrough paper, Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili proved this to be false by constructing a counterexample in the plane with an unspecified, but large, number of holes and raised the question of the minimum number of holes a counterexample can have. In this talk I will present a computer assisted counter example with 6 holes. This is joint work with Javier Gómez-Serrano and Kimberly Hou.

A computer assisted counterexample to Payne’s nodal line conjecture with few holes

April 20, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Non-uniqueness and error bounds for fluid flow

For hyperbolic systems of conservation laws in one space dimension, a general existence-uniqueness theory is now available, for entropy weak solutions with bounded variation. In several space dimensions, however, it seems unlikely that a similar theory can be achieved.

For the 2-D Euler equations, in this talk I shall discuss the simplest possible examples of Cauchy problems admitting multiple solutions. Several numerical simulations will be presented, for incompressible as well as compressible flow, indicating the existence of two distinct solutions for the same initial data. Typically, one of the solutions contains a single spiraling vortex, while the other solution contains two vortices.

Some theoretical work, aimed at validating the numerical results, will also be discussed.

Non-uniqueness and error bounds for fluid flow

April 13, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Relative equilibria for the n-body problem

We will discuss the classical problem from celestial mechanics of determining the number of relative equilibria a set of planets can display. Several already established results will be presented, as well as a new contribution (in terms of a new proof) for the restricted 4-body problem. We will discuss its possible extensions to harder instances of the general problem. This is joint work with Piotr Zgliczynski and Jordi-Lluis Figueras.

Relative equilibria for the n-body problem

April 6, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Rigorous FEM based forward in time integration of dissipative PDEs

We present the technique for computer assisted rigorous forward in time integration of problems governed by dissipative PDEs. The approach is based on the Finite Element Method. The key concepts lie in the propagation of the a priori energy estimates needed to bound the infinite dimensional remainder and in rigorous integration of differential inclusions. The technique is illustrated by the computer assisted construction of the time periodic solution for periodically forced one-dimensional Burgers equation with homogeneous Dirichlet boundary conditions. Talk is based on joint work with Piotr Zgliczyński.

Rigorous FEM based forward in time integration of dissipative PDEs

March 30, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Computer-assisted proofs for Cauchy problems of delay equations and PDEs via Chebyshev series

In this talk we introduce recent general methods to rigorously compute solutions of infinite dimensional Cauchy problems. The idea is to expand the solutions in time using Chebyshev series and use the contraction mapping theorem to construct a neighbourhood about an approximate solution which contains the exact solution of the Cauchy problem. We apply the methods to delay differential equations and to semi-linear parabolic partial differential equations.

Computer-assisted proofs for Cauchy problems of delay equations and PDEs via Chebyshev series

March 23, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Existence of traveling wave solutions for the Diffusion Poisson Coupled Model: a computer-assisted proof

In France one option under study for the storage of high-level radioactive waste is based on an underground repository. More precisely, the waste shall be confined in a glass matrix and then placed into cylindrical steel canisters. These containers shall be placed into micro-tunnels in the highly impermeable Callovo-Oxfordian claystone layer at a depth of several hundred meters. The Diffusion Poisson Coupled Model (DPCM) aims to investigate the safety of such long term repository concept by describing the corrosion processes appearing at the surface of carbon steel canisters in contact with a claystone formation. It involves drift-diffusion equations on the density of species (electrons, ferric cations and oxygen vacancies), coupled with a Poisson equation on the electrostatic potential and with moving boundary equations. So far, no theoretical results giving a precise description of the solutions, or at least under which conditions the solutions may exist, are avalaible in the literature. However, a finite volume scheme has been developed to approximate the equations of the DPCM model. In particular, it was observed numerically the existence of traveling wave solutions for the DPCM model. These solutions are defined by stationary profiles on a fixed size domain with interfaces moving at the same velocity. The main objective of this talk is to present how we apply a computer-assisted method in order to prove the existence of such traveling wave solutions for the system. This approach allows us to obtain for the first time a precise and certified description of some solutions.

This work is in collaboration with Maxime Breden and Claire Chainais-Hillairet.

Existence of traveling wave solutions for the Diffusion Poisson Coupled Model: a computer-assisted proof

March 16, 2021 from 9:00 to 10:00 (Montreal/EST time) Zoom meeting

### Computer assisted existence proof of complicated dynamics in forced delay action oscillator modelling El Nino phenomena

A computer assisted proof is presented for the existence of various periodic solutions for forced Suarez-Schopf's equation, which are delay differential equations modeling El Nino. Tight inclusions of periodic solutions are calculated through numerical verification method by utilizing a structure of Galerkin's equation for forced Suarez-Schopf's equation effectively. The existence of various periodic solutions has been proved via computer assisted proofs including various subharmonics. Especially, coexistence of several subharmonics are proved and numerical simulations are presented suggesting an appearance of chaos.

Computer assisted existence proof of complicated dynamics in forced delay action oscillator modelling El Nino phenomena

March 9, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### KAM (computer-assisted) results in Celestial Mechanics: the dissipative spin-orbit problem

The existence of invariant tori through Kolmogorov-Arnold-Moser (KAM) theory has been proven in several models of Celestial Mechanics through dedicated analytical proofs combined with computer-assisted techniques. After reviewing some of such results, obtained in conservative frameworks, we present a recent result on the existence of invariant attractors for a dissipative model: the spin-orbit problem with tidal torque. This model belongs to the class of conformally symplectic systems, which are characterized by the property that they transform the symplectic form into a multiple of itself. Finding the solution of such systems requires to add a drift parameter.

We describe a KAM theorem for conformally symplectic systems in an a-posteriori format: assuming the existence of an approximate solution, satisfying the invariance equation up to an error term - small enough with respect to explicit condition numbers, - then we can prove the existence of a solution nearby. The theorem, which does not assume that the system is close to integrable, yields an efficient algorithm to construct invariant attractors for the spin-orbit problem and it provides accurate estimates of the breakdown threshold of the invariant attractor.

This talk refers to joint works with R. Calleja, J. Gimeno, and R. de la Llave.

KAM (computer-assisted) results in Celestial Mechanics: the dissipative spin-orbit problem

March 2, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Boundary value problems and transversality in conservative systems: computer assisted proofs of connection and collision orbits

I'll discuss a framework for two-point boundary value problems in conservative systems which detects transversality, allows for the possibility of multiple changes of coordinates, and leads naturally to computer assisted proofs.  The set-up applies to dynamical problems in the level set like finding connecting orbits between hyperbolic invariant objects and collisions.  The main technical difficulty is that the conserved quantity leads to overdetermined systems of equations.  This problem can be overcome in a number of different ways, including elimination of an equation, by exploiting discrete symmetries (if any), or by introducing a new variable called an unfolding parameter.   I'll look at two common ways of defining unfolding parameters and show that they don't disrupt the transversality properties of the BVP.  I'll also illustrate some applications of this setup to computer assisted proofs of connecting orbits and collisions in the circular restricted three body problem.  This is joint work with Shane Kepley and Maciej Capinski.

Boundary value problems and transversality in conservative systems: computer assisted proofs of connection and collision orbits

February 23, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Beyond Exponential Complexity of Newton-Galerkin Validation Methods: A Polynomial-Time Newton-Picard Validation Algorithm for linear ODEs

A wide range of techniques have been developed to compute validated numerical solutions to various kind of equations (e.g., ODE, PDE, DDE) arising in computer-assisted proofs. Among them are Newton-Galerkin a posteriori validation techniques, which provide error bounds for approximate solutions by using the contraction map principle in a suitable coefficient space (e.g., Fourier or Chebyshev). More precisely, a contracting Newton-like operator is constructed by truncating and inverting the inverse Jacobian of the equation.

While these techniques were extensively used in cutting-edge works in the community, we show that they suffer from an exponential running time w.r.t. the input equation. We illustrate this shortcomings on simple linear ODEs, where a "large" parameter in the equation leads to an intractable instance for Newton-Galerkin validation algorithms.

From this observation, we build a new validation scheme, called Newton-Picard, which breaks this complexity barrier. The key idea consists in replacing the inverse Jacobian not by a finite-dimensional truncated matrix as in Newton-Galerkin, but by an integral operator with a polynomial approximation of the so-called resolvent kernel. Moreover, this method is also less basis-dependent and more suitable to be formalized in a computer proof assistant towards a fully certified implementation in the future.

Beyond Exponential Complexity of Newton-Galerkin Validation Methods: A Polynomial-Time Newton-Picard Validation Algorithm for linear ODEs

February 16, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### CRM-CAMP COLLOQUIUM: Wherefore computer assisted proofs in dynamics?

Over the past few decades the topic of computer assisted proofs in nonlinear dynamics has blossomed and is well on the way to becoming a standard part of the field. So perhaps it is worth reflecting on some high level topics. With this in mind I will discuss, from an admittedly biased personal perspective, several questions:

• Why do computer assisted proofs?
• Where do computer assisted proofs in dynamics as currently being done lie in the bigger scheme of formal proof systems?
• What new perspective about nonlinear dynamics can we extract from computer assisted proofs?
• How should we resolve the dichotomy between precision and accuracy?
• What role do computer assisted proofs have to play as we move into an era of data driven science and machine learning?

CRM-CAMP COLLOQUIUM: Wherefore computer assisted proofs in dynamics?

February 9, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Interval methods with Julia: Finding one million roots in one second

The Julia language provides a remarkably productive environment for scientific computing, with a unique combination of interactivity and speed, and is particularly suitable for defining operations on new mathematical objects, such as intervals.

I will present our free / open-source packages for interval arithmetic and interval methods (juliaintervals.github.io), written in pure Julia and comparable to state-of-the-art libraries. They use the composability coming from Julia's "multiple-dispatch"-based design and generic programming to integrate with other packages in the "ecosystem", including linear algebra, automatic differentiation, and plotting.

The foundation is IntervalArithmetic.jl , which is almost compliant with the IEEE-1788 standard. Applications currently implemented include root finding, global optimization, constraint programming, Taylor models, and validated integration of ODEs.

I will also show how Julia's facilities for parallel computing allow us to create user-defined objects on GPUs and manipulate them using the same Julia code. As an example benchmark, we find and verify one million stationary points of the two-dimensional transcendental Griewank function in under one second.

Joint work with Luis Benet (ICF-UNAM).

Interval methods with Julia: Finding one million roots in one second

February 2, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Golden mean renormalization for the almost Mathieu operator and related skew products

We renormalize SL(2,R) skew-product maps over circle rotations. Such maps arise e.g. in the spectral analysis of the Hofstadter Hamiltonian and the almost Mathieu operator. For rotations by the inverse golden mean, we prove the existence of two renormalization-periodic orbits. We conjecture that there are infinitely many such orbits, and that the associated universal constants describe local scaling properties of the Hofstadter spectrum and of the corresponding generalized eigenvectors. Some recent results on trigonometric skew-product maps will be described as well. This is joint work with Saša Kocić (UT Austin, USA).

Golden mean renormalization for the almost Mathieu operator and related skew products

January 26, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Periodic orbits in Rössler system

In a joint work with Piotr Zgliczynski, we study the Rössler system with an attracting periodic orbit, for two sets of parameters. In both cases the attractor on a Poincare section seems to be almost one-dimensional and therefore we apply the methods for two-dimensional perturbations of an interval's self-map introduced by Zgliczynski in Multidimensional perturbations of one-dimensional maps and stability of Sharkovskii ordering in 1999. We prove the existence of p-periodic orbits for almost all natural p with computer assistance: by interval Newton method and covering relations.

Periodic orbits in Rössler system

January 19, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Correct rounding for transcendental functions

On a computer, real numbers are usually represented by a finite set of numbers called floating-point numbers. When one performs an operation on these numbers, such as an evaluation by a function, one returns a floating-point number, hopefully close to the mathematical result of the operation. Ideally, the returned result should be the exact rounding of this mathematical value. If we’re only allowed a unique and fast evaluation (a constraint often met in practice), one knows how to guarantee such a quality of results for arithmetical operations like +,−,x,/ and square root but, as of today, it is still an issue when it comes to evaluate an elementary function such as cos, exp, cube root for instance. This problem, called Table Maker’s Dilemma, is actually a diophantine approximation problem. It was tackled, over the last fifteen years, by V. Lefèvre, J.M. Muller, D. Stehlé, A. Tisserand and P. Zimmermann (LIP, ÉNS Lyon and LORIA, Nancy), using tools from algorithmic number theory. Their work made it possible to partially solve this question but it remains an open problem. In this talk, I will present a joint work with Guillaume Hanrot (ÉNS Lyon, LIP, AriC) that improve on a part of the existing results.

Correct rounding for transcendental functions

January 12, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### A computer assisted proof of chaos in a delayed perturbation of chaotic ODE

We will discuss some recent developments to the Taylor method for forward in time rigorous integration of Delay Differential Equations (DDEs) with constant delays. We briefly discuss how to generalize method of the paper "Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation, Found. Comp. Math., 18 (6), 1299-1332, 2018" to incorporate multiple lags, multiple variables (systems of equations) and how to utilize "smoothing of solutions" to produce results of a far greater accuracy, especially when computing Poincaré maps between local sections. We will apply this method to validate some covering relations between carefully selected sets under Poincaré maps defined with a flow associated to a DDE. Together with standard topological arguments for compact maps it will prove existence of a chaotic dynamics, in particular the existence of infinite (countable) number of periodic orbits. The DDE under consideration is a toy example made by adding a delayed term to the Rössler ODE under parameters for which chaotic attractor exists. The delayed term is small in amplitude, but the lag time is macroscopic (not small). This is a joint work with Piotr Zgliczyński.

A computer assisted proof of chaos in a delayed perturbation of chaotic ODE

December 8, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### CRM-CAMP COLLOQUIUM: A complete proof of the Feigenbaum conjectures

In the late 1970s, Mitchell Feigenbaum discovered the universality of bifurcations in one-parameter families of maps. This universality was explained with a fixed point equation, and a flow in the space of all one-parameter families of maps. With Peter Wittwer, I showed in 1987 a rigorous proof of this phenomenon, using a "computer assisted" proof of the Feigenbaum conjectures. I will explain what are the somehow unconventional issues of this computer assisted proof, and how they are solved. Please keep in mind that this is quite old stuff, and a more modern implementation of the ideas would be much easier now than it was over 30 years ago.

CRM-CAMP COLLOQUIUM: A complete proof of the Feigenbaum conjectures

December 1, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### OPEN PROBLEMS SERIES: Defect and front dynamics: analysis and computation

This will be a personal selection of results and related open problems in dissipative spatially extended systems. I will focus on simple, sometimes universal models such as the complex Ginzburg-Landau, the Swift-Hohenberg, and extended KPP equations and attempts to describe their dynamics based on coherent structures. I will present "conceptual' analytical results, and describe gaps that rigorous computations may be able to close. Topics include invasion fronts, defects in one- and two-dimensional oscillatory media, and point defects in striped phases.

OPEN PROBLEMS SERIES: Defect and front dynamics: analysis and computation

November 24, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. The uniqueness of stationary solutions is studied in dependence of the kinematic viscosity. For some particular forcing, it is shown that uniqueness persists on some continuous branch of stationary solutions, when the viscosity becomes arbitrarily small. On the other hand, for a different forcing, a branch of symmetric solutions is shown to bifurcate, giving rise to a secondary branch of nonsymmetric stationary solutions. Furthermore, as the kinematic viscosity is varied, the branch of symmetric stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation

November 17, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Stable periodic orbits for the Mackey-Glass equation

We consider the classical Mackey-Glass delay differential equation. By letting n go to infinity in the part encapsulating the delayed term, we obtain a limiting hybrid delay equation. We investigate how periodic solutions of this limiting equation are related to periodic solutions of the original MG equation for large n. Then, we establish a procedure for constructing such periodic solutions via forward time integration. Finally, we use rigorous numerics to establish the existence of stable periodic orbits for various parameters of the MG equation. We note that some of these solutions exhibit seemingly complicated dynamics, yet they are stable periodic orbits.

Stable periodic orbits for the Mackey-Glass equation

November 10, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Recent progress in proving stability of traveling waves in the 1D Navier-Stokes equations using rigorous computations

We discuss recent progress developing and applying rigorous computation to prove stability of traveling waves in the 1D Navier-Stokes equation. In particular, we talk about rigorous computation of the Evans function, an analytic function whose zeros correspond to eigenvalues of the linearized PDE problem. Nonlinear stability results by Zumbrun and collaborators show that the underlying traveling waves are stable if there are no eigenvalues in the right half of the complex plane. Thus one may use rigorous computation of the Evans function to prove nonlinear-orbital stability of traveling waves.

Recent progress in proving stability of traveling waves in the 1D Navier-Stokes equations using rigorous computations

November 3, 2020 from 16:00 to 17:00 (Montreal/EST time) Zoom meeting

### Stability and approximation of statistical limit laws

The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. We prove stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We then apply these results to piecewise expanding maps in one and multiple dimensions. This theory can be extended to uniformly hyperbolic maps and in this setting we develop two new Fourier-analytic methods to provide the first rigorous estimates of the variance and rate function for Anosov maps.  This is joint work with Harry Crimmins.

Stability and approximation of statistical limit laws

October 27, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Equilibrium validation in models for pattern formation based on Sobolev embeddings

In this talk, I describe a method of computer-assisted proof focused on continuation of solutions depending on a parameter. These techniques are applied to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three. The functional analytic approach and techniques can be generalized to other parabolic partial differential equations. This is joint work with Thomas Wanner (George Mason University).

Equilibrium validation in models for pattern formation based on Sobolev embeddings

October 20, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Formally verified computer-assisted mathematical proofs

Proof assistants are pieces of software designed for defining formally mathematical objects, statement and proofs. In particular, such a formalization reduces the verification of proofs to a purely mechanical well-formedness check. Since the early 70s, proof assistants have been extensively used for applications in program verification, notably for security-related issues. They have also been used to verify landmark results in mathematics, including theorems with a computational proof, like the Four Colour Theorem (Appel and Haken, 1977) or Hales and Ferguson's proof of the Kepler conjecture (2005). This talk will discuss what are formalized mathematics and formal proofs, and sketch the architecture of modern proof assistants. It will also showcase a few applications in formally verified rigorous computation.

The slides of the talk are available here

October 13, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Rigorous computation of periodic solutions and Floquet multipliers in delay differential equations with time-forced discontinuities

I will present some recent work on rigorous computation of periodic solutions for delay differential equations with impulse effects. At fixed moments in time, the state of such a system is reset and solutions become discontinuous. Once a periodic solution of such a system has been computed, its Floquet spectrum can be rigorously computed by discretization of the monodromy operator (period map) and some technical error estimates. As an application, we compute a branch of periodic solutions in the pulse-harvested Hutchinson equation and examine its stability.

Rigorous computation of periodic solutions and Floquet multipliers in DDEs with time-forced discontinuities

October 6, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

In this talk, we introduce a numerical method for rigorously finding the monodromy matrix of hypergeometric differential equations. From a base point defined by fundamental solutions, we analytically continue the solution on a contour around a singular point of the differential equation using a rigorous integrator. Depending on the contour we obtain the monodromy representation of fundamental solutions, which represents the fundamental group of the equation. As an application of this method, we consider a Picard-Fuchs type hypergeometric differential equation arising from a polarized K3 surface. The monodromy matrix shows a deformation of homologically independent 2-cycles for the surface along the contour, which is regarded as a change of characterization for the K3 surface. This is joint work with Naoya Inoue (University of Tsukuba) and Toshimasa Ishige (Chiba University).

Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

September 29, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Property (T) was introduced in 1967 by Kazhdan and is an important rigidity property of groups. The most elementary way to define it is through a fixed point property: a group G has property (T) if every action of G by affine isometries on a Hilbert space has a fixed point. Property (T) has numerous applications in the form of rigidity of actions and operator algebras associated to the group, constructions of expander graphs or constructions of counterexamples to Baum-Connes-type conjectures.

In this talk I will give a brief introduction to property (T) and explain the necessary group-theoretic background in order to present a computer-assisted approach to proving property (T) by showing that the Laplacian on the group has a spectral gap. This approach allowed us show that Aut(F_n), the group of automorphisms of the free group F_n on n generators, has property (T) when n is at least 5: the case n=5 is joint work with Marek Kaluba and Narutaka Ozawa, and the case of n at least 6 is joint work with Kaluba and Dawid Kielak. Important aspects of our methods include passing from a computational result to a rigorous proof and later obtaining the result for an infinite family of groups using a single computation. I will present an overview of these arguments.

A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

September 22, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Computing and validating collisions, ejections, and homoclinics for the three body problem

Understanding connecting and collision/ejection orbits is central to the study of transport in Celestial Mechanics. The atlas algorithm combines the parameterization method with rigorous numerical techniques for solving initial value problems in order to find and validate connecting orbits. However, difficulties arise when parameterizing orbits passing near a singularity such as “near miss” homoclinics or ejection/collision orbits. In this talk we present a method of overcoming this obstacle based on rigorous Levi-Civita regularization which desingularizes the vector field near the primaries. This regularization is performed dynamically allowing invariant manifolds to be parameterized globally, even near singularities.

Computing and validating collisions, ejections, and homoclinics for the three body problem

September 15, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

We prove the existence of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE. For this, we rewrite the Kuramoto–Sivashinsky equation into a desingularized formulation near the Hopf point via a blow-up approach and we apply the radii polynomial approach to validate a solution branch of periodic solutions. Then this solution branch includes the Hopf bifurcation point.

Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

September 8, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

In this talk I will present a Computer Aided Proof of Noise Induced Order (NIO) in a model associated with the Belousov-Zhabotinsky reaction: when studying the random dynamical system with additive noise associated to the BZ map, as the noise amplitude increases the Lyapunov exponent of the model transitions from positive to negative. The proof is obtained through rigorous approximation of the stationary measure using Ulam method.

I will also show a sufficient condition for the existence of NIO in a wide family of one-dimensional examples.

[1] S. Galatolo, M. Monge, I. Nisoli "Existence of Noise Induced Order: a computer aided proof", Nonlinearity 33(9)
[2] I. Nisoli "Sufficient Conditions for Noise Induced Order in 1-dimensional systems", arXiv:2003.08422

A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

September 1, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Computer assisted proofs of Arnold Diffusion

We will present three methods that can be used for computer assisted proofs of Arnold diffusion in Hamiltonian systems. The first is the classical Melnikov method; the second is based a shadowing lemma in the setting of the scattering map theory; the last is based on topological shadowing using correctly aligned windows and cones. We will also discuss an application in the setting of the Planar Elliptic Restricted Three Body Problem.

Computer assisted proofs of Arnold Diffusion

August 25, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

In this talk, we discuss a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains, we reformulate the problem into the rigorous computation of eigenvectors of some elliptic PDEs, namely the Kolmogorov/Fokker-Planck equations describing distributions of the underlying stochastic process, and are thus able to prove that  the first Lyapunov exponent is positive for certain parameter regimes.

Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

August 18, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Rigorously validated estimation of statistical properties of expanding maps

Full-branch uniformly expanding maps and their long-time statistical quantities serve as common models for chaotic dynamics, as well as having applications to number theory. I will present an efficient method to compute important statistical quantities such as physical invariant measures, which can obtain rigorously validated bounds. To accomplish this, a Chebyshev Galerkin discretisation of transfer operators of these maps is constructed; the spectral data at the eigenvalue 1 is then approximated from this discretisation. Using this method we obtain validated estimates of Lyapunov exponents and diffusion coefficients that are accurate to over 100 decimal places. These methods may also fruitfully be extended to non-uniformly expanding maps of Pomeau-Manneville type, which have largely been altogether resistant to numerical study.

Rigorously validated estimation of statistical properties of expanding maps

August 11, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Torus knot choreographies in the N-body problem

N-body choreographies are periodic solutions to the N-body equations in which equal masses chase each other around a fixed closed curve. In this talk I will present a systematic approach for proving the existence of spatial choreographies in the gravitational body problem with the help of the digital computer. These arise from the polygonal system of bodies in a rotating frame of reference. In rotating coordinates, after exploiting the symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies.

Torus knot choreographies in the N-body problem

August 4, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE

We give a computer-assisted proof of the existence of symbolic dynamics for a certain Poincaré map in the one-dimensional Kuramoto-Sivashinsky PDE. In particular, we show the existence of infinitely many (countably) periodic orbits (POs) of arbitrary large principal periods. We provide also a study of the stability type of some POs and show the existence of a countable infinity of geometrically different homoclinic orbits to a periodic solution. The proof utilizes pure topological results (variant of the method of covering relations on compact absolute neighbourhood retracts) with rigorous integration of the PDE and the associated variational equation. This talk is based on the recent results [1,2].

[1] D. Wilczak and P. Zgliczyński. A geometric method for infinite-dimensional chaos: symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line, Journal of Differential Equations, Vol. 269 No. 10 (2020), 8509-8548.
[2] D. Wilczak and P. Zgliczyński. A rigorous C1-algorithm for integration of dissipative PDEs based on automatic differentiation and the Taylor method, in preparation.

Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE

July 28, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### A modification of Schiffer's conjecture, and a proof via finite elements

Approximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues for the Dirichlet Laplacian in certain domains. If these are to be used as part of a proof, care must be taken to ensure each step of the computation is validated and verifiable. In this talk we present a long-standing conjecture in spectral geometry, and its resolution using validated finite element computations.  Schiffer’s conjecture states that if a bounded domain Ω in R^n has any nontrivial Neumann eigenfunction which is a constant on the boundary, then Ω must be a ball. This conjecture is open. A modification of Schiffer’s conjecture is: for regular polygons of at least 5 sides, we can demonstrate the existence of a Neumann eigenfunction which does not change sign on the boundary. In this talk, we provide a recent proof using finite element calculations for the regular pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains of the triangle with mixed Dirichlet and Neumann boundary conditions. We use a learning algorithm to find and optimize this sequence of subdomains, and use non-conforming linear FEM to compute validated lower bounds for the lowest eigenvalue in each of these domains. The linear algebra is performed within interval arithmetic. This talk is based on the following paper, which is a joint work with Bartlomiej Siudeja and Ben Green at University of Oregon.

A modification of Schiffer's conjecture, and a proof via finite elements

July 21, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains

We consider the solution verification for the stationary Navier-Stokes equation over a bounded non-convex 3D domain Ω. In 1999, M.T. Nakao, et al., reported a solution existence verification example for the 2D square domain.  However, it has been a difficult problem to deal with general 2D domains and 3D domains, due to the bottleneck problem in the  a priori error estimation for the linearized NS equation. Recently, by extending the hypercircle method (Prage-Synge's theorem) to deal with the divergence-free condition in the Stokes equation, the explicit error estimation is constructed successfully based on a conforming finite element approach [arXiv:2006.02952]. Further,  we succeeded in the solution existence verification for the stationary NS equation in several nonconvex 3D domains.  In this talk, I will show the latest progress on this topic, including the rigorous estimation of the eigenvalue of Stokes operator in 3D domains.

Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains

July 14, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Uniqueness of Whitham's highest cusped wave

Whitham’s equation of shallow water waves is a non-homogeneous non-local dispersive equation. As in the case of the Stokes wave for the Euler equation, non-smooth traveling waves with greatest height between crest and trough have been shown to exist. In this talk I will discuss uniqueness of solutions to the Whitham equation and show that there exists a unique, even and periodic traveling wave of greatest height, that moreover has a convex profile between consecutive stagnation points, at which there is a cusp. Joint work with Alberto Enciso and Bruno Vergara.

Uniqueness of Whitham's highest cusped wave

July 7, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains

Many boundary value problems for semilinear elliptic partial differential equations allow very stable numerical computations of approximate solutions, but are still lacking analytical existence proofs. In this lecture, we propose a method which exploits the knowledge of a "good" numerical approximate solution, in order to provide a rigorous proof of existence of an exact solution close to the approximate one. This goal is achieved by a fixed-point argument which takes all numerical errors into account, and thus gives a mathematical proof which is not "worse" than any purely analytical one. A crucial part of the proof consists of the computation of eigenvalue bounds for the linearization of the given problem at the approximate solution. The method is used to prove existence and multiplicity statements for some specific examples, including cases where purely analytical methods had not been successful.

Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains

June 30, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

In this talk we study the nonlinear Schrödinger equation  posed on the 1-torus. Based on their numerics, Cho, Okamoto, & Shōji conjectured in their 2016 paper that: (C1) any singularity in the complex plane of time arising from inhomogeneous initial data is a branch singularity, and (C2) real initial data will exist globally in real time. If true, Conjecture 1 would suggest a strong incompatibility with the Painlevé property, a property closely associated with integrable systems. While Masuda proved (C1) in 1983 for close-to-constant initial data, a generalization to other initial data is not known. Using computer assisted proofs we establish a branch singularity in the complex plane of time for specific, large initial data which is not close-to-constant.

Concerning (C2), we demonstrate an open set of initial data which is homoclinic to the 0-homogeneous-equilibrium, proving (C2) for close-to-constant initial data. This proof is then extended to a broader class of nonlinear Schrödinger equation without gauge invariance, and then used to prove the non-existence of any real-analytic conserved quantities. Indeed, while numerical evidence suggests that most initial data is homoclinic to the 0-equilibrium, there is more than meets the eye. Using computer assisted proofs, we establish an infinite family of unstable nonhomogeneous equilibria, as well as heteroclinic orbits traveling between these nonhomogeneous equilibria and the 0-equilibrium.

An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

June 23, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### Stable periodic patterns in 3D for the Ohta-Kawasaki problem

In this talk we discuss a mathematically rigorous computational method to compare local minimizers of the Ohta-Kawasaki free energy, describing diblock copolymer melts. This energy incorporates a nonlocal term to take into account the bond between the monomers.

Working within an arbitrary space group symmetry, we explore the phase space, computing candidates both with and without experimentally observed symmetries. We validate the phase diagram, identifying regions of parameter space where different spatially periodic structures have the lowest energy. These patterns may be lamellar layers, hexagonally packed cylinders, body-centered or close-packed spheres, as well as double gyroids and 'O70' arrangements. Each computation is validated by a mathematical theorem, where we bound the truncation errors and apply a fixed point argument to establish a computer-assisted proof. The method can be applied more generally to symmetric space-time periodic solution of many partial differential equations.

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Start time Title Speaker
2021-06-29 10:00 Computer-assisted proofs for renormalisation fixed-points and eigenfunctions for period-doubling universality in maps of the interval Andrew Burbanks (University of Portsmouth, UK)
2021-06-22 10:00 Combinatorial Topological Dynamics Marian Mrozek (Jagiellonian University, Poland)
2021-06-15 10:00 Chaos in the Chua's circuit Zbigniew Galias (AGH University of Science and Technology, Poland)
2021-06-08 10:00 OPEN PROBLEMS SERIES: Some conjectures that seem difficult to prove Vladimir Sverak (University of Minnesota)
2021-06-01 10:00 CRM CAMP Colloquium: Encounters with Computer-Assisted Proofs in Early Days Charles Fefferman (Princeton University, USA)
2021-05-18 10:00 Bifurcation Points in the Ohta-Kawasaki Model Thomas Wanner (George Mason University, USA)
2021-05-11 10:00 A computer assisted proof of Wright's conjecture: counting and discounting slowly oscillating periodic solutions to a DDE Jonathan Jaquette (Boston University, USA)
2021-05-04 10:00 Rigorous numerics of blow-up solutions for autonomous ODEs Kaname Matsue (Kyushu University, Japan)
2021-04-27 10:00 A computer assisted counterexample to Payne’s nodal line conjecture with few holes Joel Dahne (Uppsala University, Sweden)
2021-04-20 10:00 Non-uniqueness and error bounds for fluid flow Alberto Bressan (Penn State University, USA)
2021-04-13 10:00 Relative equilibria for the n-body problem Warwick Tucker (Monash University, Australia)
2021-04-06 10:00 Rigorous FEM based forward in time integration of dissipative PDEs Piotr Kalita (Jagiellonian University, Poland)
2021-03-30 10:00 Computer-assisted proofs for Cauchy problems of delay equations and PDEs via Chebyshev series Jean-Philippe Lessard (McGill University, Canada)
2021-03-23 10:00 Existence of traveling wave solutions for the Diffusion Poisson Coupled Model: a computer-assisted proof Antoine Zurek (Technische Universität Wien, Austria)
2021-03-16 09:00 Computer assisted existence proof of complicated dynamics in forced delay action oscillator modelling El Nino phenomena Shin'ichi Oishi (Waseda University, Japan)
2021-03-09 10:00 KAM (computer-assisted) results in Celestial Mechanics: the dissipative spin-orbit problem Alessandra Celletti (University of Rome Tor Vergata, Italy)
2021-03-02 10:00 Boundary value problems and transversality in conservative systems: computer assisted proofs of connection and collision orbits Jason D. Mireles James (Florida Atlantic University, USA)
2021-02-23 10:00 Beyond Exponential Complexity of Newton-Galerkin Validation Methods: A Polynomial-Time Newton-Picard Validation Algorithm for linear ODEs Florent Bréhard (Université de Lille, France)
2021-02-16 10:00 CRM-CAMP COLLOQUIUM: Wherefore computer assisted proofs in dynamics? Konstantin Mischaikow (Rutgers University, USA)
2021-02-09 10:00 Interval methods with Julia: Finding one million roots in one second David Sanders (Universidad Nacional Autonoma de Mexico, Mexico)
2021-02-02 10:00 Golden mean renormalization for the almost Mathieu operator and related skew products Hans Koch (University of Texas at Austin, USA)
2021-01-26 10:00 Periodic orbits in Rössler system Anna Gierzkiewicz (Agriculture University in Krakow, Poland)
2021-01-19 10:00 Correct rounding for transcendental functions Nicolas Brisebarre (CNRS, ENS Lyon, France)
2021-01-12 10:00 A computer assisted proof of chaos in a delayed perturbation of chaotic ODE Robert Szczelina (Jagiellonian University, Poland)
2020-12-08 10:00 CRM-CAMP COLLOQUIUM: A complete proof of the Feigenbaum conjectures Jean-Pierre Eckmann (University of Geneva, Switzerland)
2020-12-01 10:00 OPEN PROBLEMS SERIES: Defect and front dynamics: analysis and computation Arnd Scheel (University of Minnesota, USA)
2020-11-24 10:00 Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation Gianni Arioli (Politecnico di Milano, Italy)
2020-11-17 10:00 Stable periodic orbits for the Mackey-Glass equation Ferenc Bartha (University of Szeged, Hungary)
2020-11-10 10:00 Recent progress in proving stability of traveling waves in the 1D Navier-Stokes equations using rigorous computations Blake Barker (Brigham Young University, USA)
2020-11-03 16:00 Stability and approximation of statistical limit laws Gary Froyland (UNSW Sydney, Australia)
2020-10-27 10:00 Equilibrium validation in models for pattern formation based on Sobolev embeddings Evelyn Sander (George Mason University, USA)
2020-10-20 10:00 Formally verified computer-assisted mathematical proofs Assia Mahboubi (Inria, France & VU Amsterdam, Netherlands)
2020-10-13 10:00 Rigorous computation of periodic solutions and Floquet multipliers in delay differential equations with time-forced discontinuities Kevin Church (McGill University, Canada)
2020-10-06 10:00 Computer-assisted proofs for finding the monodromy of hypergeometric differential equations Akitoshi Takayasu (University of Tsukuba, Japan)
2020-09-29 10:00 A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups Piotr Nowak (Polish Academy of Sciences, Poland)
2020-09-22 10:00 Computing and validating collisions, ejections, and homoclinics for the three body problem Shane Kepley (Rutgers University, USA)
2020-09-15 10:00 Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE Elena Queirolo (Rutgers University, USA)
2020-09-08 10:00 A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon Isaia Nisoli (Universidade Federal do Rio de Janeiro, Brazil)
2020-09-01 10:00 Computer assisted proofs of Arnold Diffusion Maciej Capiński (AGH University of Science and Technology, Poland)
2020-08-25 10:00 Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems Maxime Breden (École Polytechnique, France), Maximilian Engel (Freie Universität Berlin, Germany)
2020-08-18 10:00 Rigorously validated estimation of statistical properties of expanding maps Caroline Wormell (University of Sydney, Australia)
2020-08-11 10:00 Torus knot choreographies in the N-body problem Renato Calleja (Universidad Nacional Autonoma de Mexico, Mexico)
2020-08-04 10:00 Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE Daniel Wilczak (Jagiellonian University, Poland)
2020-07-28 10:00 A modification of Schiffer's conjecture, and a proof via finite elements Nilima Nigam (Simon Fraser University, Canada)
2020-07-21 10:00 Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains Xuefeng Liu (Niigata University, Japan)
2020-07-14 10:00 Uniqueness of Whitham's highest cusped wave Javier Gómez-Serrano (Brown University, USA & University of Barcelona, Spain)
2020-07-07 10:00 Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains Michael Plum (Karlsruhe Institute of Technology, Germany)
2020-06-30 10:00 An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance Jonathan Jaquette (Boston University, USA)
2020-06-23 10:00 Stable periodic patterns in 3D for the Ohta-Kawasaki problem Jan Bouwe van den Berg (VU Amsterdam, Netherlands)
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# Register

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# List of registered participants

Currently, there are 299 registered participants.

# Videos

June 29, 2021

Andrew Burbanks

June 22, 2021

Marian Mrozek

June 15, 2021

Zbigniew Galias

June 8, 2021

June 1, 2021

### CRM CAMP Colloquium: Encounters with Computer-Assisted Proofs in Early Days

Charles Fefferman

May 18, 2021

Thomas Wanner

May 11, 2021

### A computer assisted proof of Wright's conjecture: counting and discounting slowly oscillating periodic solutions to a DDE

Jonathan Jaquette

May 4, 2021

Kaname Matsue

April 27, 2021

Joel Dahne

April 20, 2021

Alberto Bressan

April 13, 2021

Warwick Tucker

April 6, 2021

Piotr Kalita

March 30, 2021

### Computer-assisted proofs for Cauchy problems of delay equations and PDEs via Chebyshev series

Jean-Philippe Lessard

March 23, 2021

Antoine Zurek

March 16, 2021

Shin'ichi Oishi

March 9, 2021

### KAM (computer-assisted) results in Celestial Mechanics: the dissipative spin-orbit problem

Alessandra Celletti

March 2, 2021

### Boundary value problems and transversality in conservative systems: computer assisted proofs of connection and collision orbits

Jason D. Mireles James

February 23, 2021

### Beyond Exponential Complexity of Newton-Galerkin Validation Methods: A Polynomial-Time Newton-Picard Validation Algorithm for linear ODEs

Florent Bréhard

February 16, 2021

### CRM-CAMP COLLOQUIUM: Wherefore computer assisted proofs in dynamics?

Konstantin Mischaikow

February 9, 2021

David Sanders

February 2, 2021

Hans Koch

January 26, 2021

### Periodic orbits in Rössler system

Anna Gierzkiewicz

January 19, 2021

### Correct rounding for transcendental functions

Nicolas Brisebarre

January 12, 2021

Robert Szczelina

December 8, 2020

### CRM-CAMP COLLOQUIUM: A complete proof of the Feigenbaum conjectures

Jean-Pierre Eckmann

December 3, 2020

Olivier Hénot

December 3, 2020

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs for a nonlinear Laplace-Beltrami equation on the sphere

Gabriel William Duchesne

December 3, 2020

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Periodic orbit for Brusselator system with diffusion

Jakub Banaśkiewicz

December 3, 2020

Maxime Murray

December 3, 2020

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs of two-dimensional attracting invariant tori for ODEs

Emmanuel Fleurantin

December 3, 2020

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted existence proofs for Navier-Stokes equations on an unbounded strip with obstacle

Jonathan Wunderlich

December 3, 2020

Joel Dahne

December 1, 2020

### OPEN PROBLEMS SERIES: Defect and front dynamics: analysis and computation

Arnd Scheel

November 24, 2020

### Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation

Gianni Arioli

November 17, 2020

### Stable periodic orbits for the Mackey-Glass equation

Ferenc Bartha

November 10, 2020

Blake Barker

November 3, 2020

Gary Froyland

October 27, 2020

Evelyn Sander

October 13, 2020

Kevin Church

October 6, 2020

### Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

Akitoshi Takayasu

September 29, 2020

### A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Piotr Nowak

September 22, 2020

### Computing and validating collisions, ejections, and homoclinics for the three body problem

Shane Kepley

September 15, 2020

### Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

Elena Queirolo

September 8, 2020

### A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

Isaia Nisoli

September 1, 2020

Maciej Capiński

August 25, 2020

### Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

Maxime Breden, Maximilian Engel

August 18, 2020

Caroline Wormell

August 11, 2020

Renato Calleja

August 4, 2020

Daniel Wilczak

July 28, 2020

Nilima Nigam

July 21, 2020

Xuefeng Liu

July 14, 2020

### Uniqueness of Whitham's highest cusped wave

Javier Gómez-Serrano

July 7, 2020

Michael Plum

June 30, 2020

### An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

Jonathan Jaquette

June 23, 2020

### Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg

# Open Problems Series

This is a series of talks focusing on either open problems concerning techniques of computer-assisted proof, or more broadly open problem in mathematics where the speaker believes there could be a computer-assisted solution. Talks range from 5 minutes to an hour, and can be proposed at any level. When an open problem is solved, or when substantial progress is made, we provide citation and links to the relevant work.

Stay up-to-date with the group's activities and receive invites to the seminars.

Register

# Past Open Problems

June 8, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### OPEN PROBLEMS SERIES: Some conjectures that seem difficult to prove

In many cases, numerics and/or heuristics provide compelling evidence for statements that we have trouble proving rigorously. I will discuss some examples, mostly inspired by fluid flows.

OPEN PROBLEMS SERIES: Some conjectures that seem difficult to prove

December 1, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### OPEN PROBLEMS SERIES: Defect and front dynamics: analysis and computation

This will be a personal selection of results and related open problems in dissipative spatially extended systems. I will focus on simple, sometimes universal models such as the complex Ginzburg-Landau, the Swift-Hohenberg, and extended KPP equations and attempts to describe their dynamics based on coherent structures. I will present "conceptual' analytical results, and describe gaps that rigorous computations may be able to close. Topics include invasion fronts, defects in one- and two-dimensional oscillatory media, and point defects in striped phases.

OPEN PROBLEMS SERIES: Defect and front dynamics: analysis and computation

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# Other Scientific Activities

Here, the different scientific activities related to the research group will be grouped together. First, the upcoming activities, then the past activities.

# Past Scientific Activities

February 16, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

### CRM-CAMP COLLOQUIUM: Wherefore computer assisted proofs in dynamics?

Over the past few decades the topic of computer assisted proofs in nonlinear dynamics has blossomed and is well on the way to becoming a standard part of the field. So perhaps it is worth reflecting on some high level topics. With this in mind I will discuss, from an admittedly biased personal perspective, several questions:

• Why do computer assisted proofs?
• Where do computer assisted proofs in dynamics as currently being done lie in the bigger scheme of formal proof systems?
• What new perspective about nonlinear dynamics can we extract from computer assisted proofs?
• How should we resolve the dichotomy between precision and accuracy?
• What role do computer assisted proofs have to play as we move into an era of data driven science and machine learning?

CRM-CAMP COLLOQUIUM: Wherefore computer assisted proofs in dynamics?

December 3, 2020 from 12:00 to 12:15 (Montreal/EST time) Zoom meeting

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Rigorous computation of the unstable manifold for equilibria of delay differential equations

We will review the parameterization method to obtain the unstable manifold of equilibria of Delay Differential Equations. Then, we will discuss how to compute rigorous error bounds for this parameterization.

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Rigorous computation of the unstable manifold for equilibria of delay differential equations

December 3, 2020 from 11:45 to 12:00 (Montreal/EST time) Zoom meeting

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Periodic orbit for Brusselator system with diffusion

We will present numerical evidence for the existence of periodic solutions to a one-dimensional Brusselator system with diffusion and Dirichlet boundary conditions. Then we discuss a plan for proving their existence by a rigorous integration of differential inclusion corresponding to the first modes of the Galerkin projection and dissipative estimations on further modes.

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Periodic orbit for Brusselator system with diffusion

December 3, 2020 from 11:15 to 11:30 (Montreal/EST time) Zoom meeting

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Parameterized invariant manifold and applications in celestial mechanics

The parameterization method is a well-known framework with proven value to parameterize hyperbolic manifolds attached to periodic solutions of ordinary differential equations. Using a Taylor expansion, one can rewrite the computation of the manifold into a recursive system of linear differential equations describing the coefficients. I will discuss this approach and how to obtain an interval enclosure of the truncated solution to the system. I will then show how validated manifolds are used to compute cycle-to-cycle connections in the case of the circular restricted three-body problem and Hill's four-body problem.

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Parameterized invariant manifold and applications in celestial mechanics

December 3, 2020 from 11:00 to 11:15 (Montreal/EST time) Zoom meeting

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs for a nonlinear Laplace-Beltrami equation on the sphere

We prove the existence and local uniqueness of radially symmetric solutions of nonlinear Laplace-Beltrami equation on the sphere by using the Radii Polynomial Theorem on Banach spaces with a combination of Taylor and Chebyshev coefficients of the solutions.

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs for a nonlinear Laplace-Beltrami equation on the sphere

December 3, 2020 from 10:45 to 11:00 (Montreal/EST time) Zoom meeting

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs of two-dimensional attracting invariant tori for ODEs

We study the existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We consider separately two important cases of rotational and resonant tori for which we describe how we apply our methods. This is a joint work with Maciej Capinski and J.D. Mireles-James.

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs of two-dimensional attracting invariant tori for ODEs

December 3, 2020 from 10:15 to 10:30 (Montreal/EST time) Zoom meeting

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted existence proofs for Navier-Stokes equations on an unbounded strip with obstacle

The incompressible stationary 2D Navier-Stokes equations are considered on an unbounded strip domain with a compact obstacle. In order to get existence and error bounds (in a Sobolev space) for a solution, an approximate solution (using divergence-free finite elements), a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution are computed. For the latter, bounds for the essential spectrum and for eigenvalues play a crucial role, especially for the eigenvalues “close to” zero. Note that, on an unbounded domain, the only possible method for computing the desired norm bound appears to be via eigenvalue bounds. In this way, the first rigorous existence proof for the Navier-Stokes problem under consideration is obtained.

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted existence proofs for Navier-Stokes equations on an unbounded strip with obstacle

December 3, 2020 from 10:00 to 10:15 (Montreal/EST time) Zoom meeting

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computation of tight enclosures for Laplacian eigenvalues

Recently, there has been interest in high-precision approximations of the fundamental eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We present computations of improved and rigorous enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue can be certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners we handle by combining expansions from all singular corners and an expansion from an interior point.

CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computation of tight enclosures for Laplacian eigenvalues

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The CRM CAMP seminar series explores the interplay between scientific computing and rigorous mathematical analysis, spotlighting research in areas like dynamical systems theory and nonlinear analysis. The field has developed rapidly over the last several decades and, with researchers spread around the globe, there is a growing need for a regular forum to share results, pose interesting questions, and discuss new directions. This project is envisioned as a kind of online community center for weekly gatherings, as well as a repository for educational materials. In addition to the weekly lecture series the program also serves as a mechanism for organizing workshops, tutorials, and other scientific activities. We hope that, by increasing the visibility of this research, the project will stimulate collaborations across existing groups and between our community and mathematicians working in other areas.

# Biography

Jean-Philippe Lessard is an associate professor at McGill University since 2017. He obtained his Ph.D. from Georgia Tech in 2007 under the supervision of Konstantin Mischaikow. He spent some time as a postdoctoral researcher at Rutgers University, at VU University Amsterdam, was awarded a fellowship from the IAS in Princeton and was a group leader at the Basque Center for Applied Mathematics. He then became a professor at Laval University, where he stayed for six years. In 2016, he was awarded the CAIMS/PIMS Early Career Award in Applied Mathematics and is currently CRM’s deputy director of scientific programs. In his research, he combines numerical analysis, topology and functional analysis to study finite and infinite dimensional dynamical systems.

# Videos

March 30, 2021

### Computer-assisted proofs for Cauchy problems of delay equations and PDEs via Chebyshev series

Jean-Philippe Lessard

# Biography

Jason D. Mireles James received his Ph.D. from the University of Texas at Austin in 2009, where he worked with Rafael de la Llave. He moved to Rutgers University where he was first a postdoc from 2010 to 2011, and then a Hill Assistant Professor in the Mathematics Department from 2011-2014. During this time, he worked closely with the group of Konstantin Mischaikow. In 2014 he joined the Department of Mathematics at Florida Atlantic University, where he currently holds the rank of associate professor. His research focuses on problems in nonlinear analysis, drawing on tools from computational mathematics, approximation theory, and functional analysis.

# Videos

March 2, 2021

### Boundary value problems and transversality in conservative systems: computer assisted proofs of connection and collision orbits

Jason D. Mireles James

# Biography

Jan Bouwe van den Berg is a full professor at Vrije Universiteit Amsterdam since 2007. He obtained his Ph.D. from Leiden University in 2000 under the supervision of Bert Peletier. He spent two years as a postdoc in Nottingham and has held visiting positions at Simons Fraser University and at McGill. He was awarded an NWO-Vici grant in 2012 and he was a CRM-Simons visiting professor in 2019. Jan Bouwe’s research revolves around dynamical systems and nonlinear partial differential equations, where he use techniques ranging from topological and variational analysis to (rigorous) computational methods to study the dynamics of patterns.

# Videos

June 23, 2020

### Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg

# Books

Warwick Tucker

### Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations

Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe

### Rigorous Numerics in Dynamics

Jean-Philippe Lessard, Jan Bouwe van den Berg

# Survey articles

SeMA, 76, pages 459–484, 2019

### Computer-assisted proofs in PDE: a survey

Javier Gómez-Serrano

Notices of the American Mathematical Society, Volume 62 (9), pages 1057-1061, 2015

### Rigorous Numerics in Dynamics

Jean-Philippe Lessard, Jan Bouwe van den Berg

Acta Numerica, Volume 19, pages 287-449, 2010

### Verification methods: rigorous results using floating-point arithmetic

Siegfried M. Rump

SIAM Review, Volume 38 (4), pages 565-604, 1996

### Computer-assisted proofs in analysis and programming in logic: a case study

Alain Schenkel, Peter Wittwer

# Schools

August 1, 2018 https://mym.iimas.unam.mx/renato/curso.html

### Computer-Assisted Proofs in Nonlinear Dynamics

Jason D. Mireles James, Jean-Philippe Lessard

The main question addressed in this course is: suppose we have computed a good numerical approximation of a solution of nonlinear equation -- can we establish the existence of a true solution nearby? Combining tools from functional analysis, complex analysis, numerical analysis, and interval computing, we see that for many of the problems mentioned above the answer is yes. A broad and example driven overview of the field of validated numerics is given.

June 29, 2021

Andrew Burbanks

June 22, 2021

Marian Mrozek

June 15, 2021

Zbigniew Galias

June 8, 2021

June 1, 2021

### CRM CAMP Colloquium: Encounters with Computer-Assisted Proofs in Early Days

Charles Fefferman

May 18, 2021

Thomas Wanner

May 11, 2021

### A computer assisted proof of Wright's conjecture: counting and discounting slowly oscillating periodic solutions to a DDE

Jonathan Jaquette

May 4, 2021

Kaname Matsue

April 27, 2021

Joel Dahne

April 20, 2021

Alberto Bressan

April 13, 2021

Warwick Tucker

April 6, 2021

Piotr Kalita

March 30, 2021

### Computer-assisted proofs for Cauchy problems of delay equations and PDEs via Chebyshev series

Jean-Philippe Lessard

March 23, 2021

Antoine Zurek

March 16, 2021

Shin'ichi Oishi

March 9, 2021

### KAM (computer-assisted) results in Celestial Mechanics: the dissipative spin-orbit problem

Alessandra Celletti

March 2, 2021

### Boundary value problems and transversality in conservative systems: computer assisted proofs of connection and collision orbits

Jason D. Mireles James

February 23, 2021

### Beyond Exponential Complexity of Newton-Galerkin Validation Methods: A Polynomial-Time Newton-Picard Validation Algorithm for linear ODEs

Florent Bréhard

February 16, 2021

### CRM-CAMP COLLOQUIUM: Wherefore computer assisted proofs in dynamics?

Konstantin Mischaikow

February 9, 2021

David Sanders

February 2, 2021

Hans Koch

January 26, 2021

### Periodic orbits in Rössler system

Anna Gierzkiewicz

January 19, 2021

### Correct rounding for transcendental functions

Nicolas Brisebarre

January 12, 2021

Robert Szczelina

December 8, 2020

### CRM-CAMP COLLOQUIUM: A complete proof of the Feigenbaum conjectures

Jean-Pierre Eckmann

December 3, 2020

Olivier Hénot

December 3, 2020

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs for a nonlinear Laplace-Beltrami equation on the sphere

Gabriel William Duchesne

December 3, 2020

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Periodic orbit for Brusselator system with diffusion

Jakub Banaśkiewicz

December 3, 2020

Maxime Murray

December 3, 2020

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted proofs of two-dimensional attracting invariant tori for ODEs

Emmanuel Fleurantin

December 3, 2020

### CRM-CAMP SPOTLIGHT ON GRADUATE RESEARCH: Computer-assisted existence proofs for Navier-Stokes equations on an unbounded strip with obstacle

Jonathan Wunderlich

December 3, 2020

Joel Dahne

December 1, 2020

### OPEN PROBLEMS SERIES: Defect and front dynamics: analysis and computation

Arnd Scheel

November 24, 2020

### Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation

Gianni Arioli

November 17, 2020

### Stable periodic orbits for the Mackey-Glass equation

Ferenc Bartha

November 10, 2020

Blake Barker

November 3, 2020

Gary Froyland

October 27, 2020

Evelyn Sander

October 13, 2020

Kevin Church

October 6, 2020

### Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

Akitoshi Takayasu

September 29, 2020

### A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Piotr Nowak

September 22, 2020

### Computing and validating collisions, ejections, and homoclinics for the three body problem

Shane Kepley

September 15, 2020

### Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

Elena Queirolo

September 8, 2020

### A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

Isaia Nisoli

September 1, 2020

Maciej Capiński

August 25, 2020

### Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

Maxime Breden, Maximilian Engel

August 18, 2020

Caroline Wormell

August 11, 2020

Renato Calleja

August 4, 2020

Daniel Wilczak

July 28, 2020

Nilima Nigam

July 21, 2020

Xuefeng Liu

July 14, 2020

### Uniqueness of Whitham's highest cusped wave

Javier Gómez-Serrano

July 7, 2020

Michael Plum

June 30, 2020

### An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

Jonathan Jaquette

June 23, 2020

### Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg