### Workshop B1 - Computational Dynamics

**Organizers:** Àngel Jorba (Universitat de Barcelona, Spain) - Hiroshi Kokubu (Kyoto University, Japan) - Warwick Tucker (Uppsala University, Sweden)

## Talks

July 13, 14:30 ~ 15:20 - Room B1

## From normal forms to KAM theory, from space debris to the rotation of the Moon

### Alessandra Celletti

### University of Rome Tor Vergata, Italy - celletti@mat.uniroma2.it

Celestial Mechanics is a test-bench for many theories of Dynamical Systems, most notably perturbation theory and KAM theory. Realistic results in concrete applications can be obtained through an accurate modeling and an appropriate study of the dynamics, which often requires a heavy computational effort. After a general discussion on perturbation theory and KAM theorem, I will consider two examples of implementations of normal forms and KAM theory in Celestial Mechanics. The first one concerns the dynamics of space debris, which can be succesfully studied through averaging theory and normal forms computations. The second example analyzes the rotation of the Moon, whose stability can be investigated through a computer-assisted implementation of KAM theory. Perturbation theory and KAM theory can be used also to investigate dissipative systems. In this context, they can give interesting results on the dynamics of space debris at low altitude and possibly on the evolution of the Moon toward its present synchronous rotation.

Joint work with Renato Calleja (National Autonomous University of Mexico, Mexico), Christos Efthymiopoulos (Academy of Athens, Greece), Fabien Gachet (University of Rome Tor Vergata, Italy), Catalin Gales (Al. I. Cuza University, Romania), Rafael de la Llave (Georgia Institute of Technology, USA) and Giuseppe Pucacco (University of Rome Tor Vergata, Italy).

July 13, 15:30 ~ 16:00 - Room B1

## Numerical algorithms and a-posteriori verification of periodic orbits of the Kuramoto-Sivashinsky equation.

### Jordi-Lluís Figueras

### Uppsala Univesrity, Sweden - figueras@math.uu.se

In this talk we will present a numerical algorithm for the computation of (hyperbolic) periodic orbits of the 1-D Kuramoto-Sivashinsky equation \[ u_t+\nu u_{xxxx}+u_{xx}+u u_x = 0, \] with $\nu>0$.

This numerical algorithm consists on applying a suitable quasi-Newton scheme. In order to do this, we need to rewrite the invariance equation that must satisfy a periodic orbit in a form that its linearization around an approximate solution is a bounded operator. We will also show how this methodology can be used to compute a-posteriori estimates of the errors of the solutions computed, leading to the rigorous verification of the existence of the periodic orbit.

If time permits, we will finish showing some numerical outputs of the algorithms presented along the talk.

Joint work with Rafael de la Llave..

July 13, 16:00 ~ 16:30 - Room B1

## Topological changes in slow-fast systems: chaotic neuron models

### Roberto Barrio

### University of Zaragoza, Spain - rbarrio@unizar.es

The wide-range assessment of brain dynamics is one of the pivotal challenges of this century. To understand how an incredibly sophisticated system such as the brain functions dynamically, it is imperative to study the dynamics of its constitutive elements -- neurons. Therefore, the design of mathematical models for neurons has arisen as a trending topic in science for a few decades. Mathematical neuron models are examples of fast-slow systems, and they exhibit several typical behaviours and bifurcations, like chaotic dynamics, spike-adding bifurcations and so on.

Two key open questions are how the chaotic behavior is organized [1, 2] and how spike-adding bifurcations influence chaotic behavior. In this talk we show how the orbit-flip (OF) codimension-2 bifurcation points, placed in homoclinic bifurcation curves and related with the spike-adding bifurcations [1, 2, 3], originate countable pencils of period-doubling and saddle-node (of limit cycles) bifurcation lines, but also of symbolic-flip bifurcations [4]. These bifurcations appear interlaced and generate the different symbolic sequences of periodic orbits, constituting the skeleton of the different chaotic attractors and determining their topological structure [4]. The study of the changes in the chaotic invariants is done by analyzing the gradual change in the spectrum of periodic orbits embedded in the invariant, and the onion-like structure in parameter space can be understood directly in terms of symbolic dynamics. The use of several numerical techniques, as continuation techniques, Lyapunov exponents, detection of unstable periodic orbits foliated to the chaotic invariants, template analysis and so on, has played a relevant role in the complete analysis of the problem [4].

[1] Barrio R., Martinez M. A. , Serrano S., Shilnikov A. (2014) Macro and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons. Chaos 24(2):023128.

[2] Barrio R., Lefranc M., Martinez M. A., Serrano S. (2015) Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models. EPL 109:20002.

[3] Barrio R., Ibanez S., Perez L. (2017) Hindmarsh-Rose model: close and far to the singular limit. Physics Letters A 381(6), 597–603.

[4] Barrio R., Lefranc M., Serrano S. (2017). Topological changes in chaotic neuron models. Preprint.

Joint work with Marc Lefranc (Universite Lille I, France), M. Angeles Martinez (University of Zaragoza, Spain) and Sergio Serrano (University of Zaragoza, Spain).

July 13, 17:00 ~ 17:30 - Room B1

## Accurate measurement of Arnold diffusion using computer-algebraic optimal normal forms

### Christos Efthymiopoulos

### Research Center for Astronomy and Applied Mathematics, Academy of Athens, Greece - cefthim@academyofathens.gr

Abstract: the numerical measurement of Arnold diffusion, i.e., the slow drift of weakly chaotic orbits along resonances in nearly-integrable systems, has proven to be a particularly difficult task for computational dynamics. The talk will show how such measurement can be practically realized even for tiny values of the perturbation parameter, showing real examples in a model introduced in the literature some years ago by Froeschl\'{e} et al. (2000). Our technique combines three elements: i) a computer-algebraic determination of the resonant normal form which is optimal in the sense of minimizing the corresponding remainder. ii) Use of the normalizing transformation in order to remove all `deformation' effects. This allows to clearly see the diffusion in `clean' canonical variables, devoid of deformation noise. iii) A Melnikov-type determination of the most important remainder terms which drive the diffusion. In step (iii) we implement a stationary or quasi-stationary phase approach for estimating the contribution of each term to the chaotic jumps of the action variables which take place along every individual homoclinic loop. This allows, in turn, to quantify the time evolution of these jumps, in excellent agreement with numerical results. Finally, the same technique allows to visualize Arnold diffusion, i.e., to show how the diffusion proceeds along the resonance in a sequence of consecutive, in time, homoclinic loops.

Joint work with M. Guzzo (Dipt. di Matematica, Universit\`a degli Studi di Padova) and R.I. Paez (RCAAM, Academy of Athens).

July 13, 17:30 ~ 18:00 - Room B1

## Splitting of separatrices in a Hamiltonian-Hopf bifurcation under periodic forcing

### Carles Simó

### Universitat de Barcelona, Spain - carles@maia.ub.es

We consider a Hamiltonian system given by a suitable truncation of the normal form of the Hamiltonian-Hopf bifurcation plus a concrete periodic non-autonomous perturbation. The goal is to study the behavior of the splitting of the 2-dimensional separatrices. Theoretical results are compared with direct computations of the invariant manifolds. An analysis of the associated Poincar\'e-Melnikov integral provides a description of the sequence of parameters corresponding to changes on the dominant harmonics of the splitting function.

Joint work with Ernest Fontich (Universitat de Barcelona) and Arturo Vieiro (Universitat de Barcelona).

July 14, 14:30 ~ 15:00 - Room B1

## Parameterization Methods for computing Normally Hyperbolic Invariant Manifolds

### Àlex Haro

### Universitat de Barcelona, Spain - alex@maia.ub.es

We present a methodology for the computation of normally hyperbolic invariant tori in families of dynamical systems. The application of the parameterization method leads to solving invariance equations for which we use Newton-like method adapted to the dynamics and the geometry of the invariant manifold and its invariant bundles. The method computes the NHIT and its internal dynamics. We apply the method to several examples, and explore some mechanisms of breakdown of invariant tori. This is a joint work with Marta Canadell.

July 14, 15:00 ~ 15:30 - Room B1

## On parameter loci of the Hénon family

### Yutaka Ishii

### Kyushu University, Japan - yutaka@math.kyushu-u.ac.jp

We characterize the hyperbolic horseshoe locus and the maximal entropy locus of the Hénon family defined on $\mathbb{R}^2$. More specifically, we show that (i) the two parameter loci are both connected and simply connected, (ii) the closure of the hyperbolic horseshoe locus coincides with the maximal entropy locus, and (iii) their boundaries are identical and piecewise real analytic with two analytic pieces. The strategy of our proof is first to extend the dynamical and the parameter spaces over $\mathbb{C}$, investigate their complex dynamical and complex analytic properties, and then reduce them to obtain the conclusion over $\mathbb{R}$. We also employ interval arithmetic together with some numerical algorithms such as set-oriented computations and the interval Krawczyk method to verify certain numerical criteria which imply analytic, combinatorial and dynamical consequences.

Joint work with Zin Arai (Chubu University, Japan).

July 14, 15:30 ~ 16:00 - Room B1

## Numerical study of wide periodic windows for the quadratic map

### Zbigniew Galias

### AGH University of Science and Technology, Poland - galias@agh.edu.pl

Periodic windows for the map $f(x)=ax(1-x)$ are studied numerically.

An efficient method to find very accurate rigorous bounds for the endpoints of the periodic window with a given symbol sequence is presented. The method is based on applying the interval Newton method to find positions of bifurcation points of corresponding periodic orbits. The method is capable to handle periodic windows with periods up to several millions. We discuss how to select an initial point for the Newton method to improve the convergence speed. The method is applied to find all $1\,966\,957\,258$ periodic windows with periods $2\leq p\leq 36$ and show that the total width of these windows is above $0.611834003131$. Positions of periodic windows' endpoints are found with the accuracy of more than 60 decimal digits.

A heuristic algorithm to locate wide periodic windows based on the results obtained for periodic windows with low periods is described. Periodic windows are classified as primary windows and period-tupling windows. Candidates for symbol sequences corresponding to wide primary windows are constructed iteratively from shorter symbol sequences by insertion of a single symbol and by substitution of a single symbol by two-symbol sequences. Symbol sequences corresponding to wide period-tupling windows are constructed iteratively from symbol sequences of primary windows and period-tupling windows found previously. The algorithm is used to find the majority of wide periodic windows with periods $p\geq 37$.

From the results concerning periodic windows found it follows that the measure of the set of regular parameters in the interval $[3,4]$ is above $0.613960137$. Using these results, we estimate that the true value of the measure of the set of regular parameters is close to $0.6139603$.

July 14, 16:00 ~ 16:30 - Room B1

## Spectral stability for the wave equation with periodic forcing

### Hans Koch

### The University of Texas at Austin, USA - koch@math.utexas.edu

We consider the spectral stability problem for Floquet-type sytems such as the wave equation with periodic forcing. Our approach is based on a comparison with finite-dimensional approximations. Specific results are obtained for a system where the forcing is due to a coupling between the wave equation and a time-period solution of a nonlinear beam equation. We prove (spectral) stability for some period and instability for another. The finite-dimensional approximations are controlled via computer-assisted estimates.

Joint work with Gianni Arioli (Politecnico di Milano, Italy).

July 14, 17:00 ~ 17:30 - Room B1

## Order theory and Conley’s Connection Matrices

### Konstantin Mischaikow

### Rutgers University, USA - mischaik@math.rutgers.edu

Conley’s decomposition theorem indicates that given any compact invariant set the dynamics off of the associated chain recurrent set is gradient-like. Furthermore, the chain recurrent set can by characterized in terms of the set of attractors and their dual repellers. However, in general invariant sets are not computable and their structure is sensitive to variation in parameters.

In this talk I will focus on efforts to develop a computationally robust theory by replacing invariant sets as the primary object of focus with order theoretic objects such as lattices and posets. In particular, I will discuss recent work on lattices of attracting neighborhoods and associated Morse tilings as a representative of the gradient like structure of dynamics, and attempts to develop an efficient computational framework for Conley’s connection matrix to rigorously identify the structure of the gradient like invariant dynamics.

Joint work with Shaun Harker (Rutgers University, USA), William Kalies (Florida Atlantic University, USA), Kelly Spendlove (Rutgers University, USA) and Robert Vandervorst (VU Amsterdam, Netherlands).

July 14, 17:30 ~ 18:00 - Room B1

## Symbolic dynamics for Kuramoto-Sivashinsky PDE on the line --- a computer-assisted proof

### Piotr Zgliczy\'nski

### Jagiellonian University, Krakow, Poland - umzglicz@cyf-kr.edu.pl

The Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter $\nu=0.1212$ is considered. We give a computer-assisted proof the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods.

The proof is based on the covering relations, the apparent existence of trannsversal heteroclinic connections between two periodic orbits and a new algorithm on rigorous intagration of dissipative PDEs based on the automatic differentiation.

Joint work with Daniel Wilczak (Jagiellonian University, Krakow, Poland).

July 15, 14:30 ~ 15:20 - Room 111

## Validated numerics for robust space mission design

### Mioara Joldes

### LAAS-CNRS, France - joldes@laas.fr

In this talk we overview several symbolic-numeric algorithms and software tools developed for robust space mission design. An application is the computation of validated impulsive optimal control for the rendezvous problem of spacecrafts, assuming a linear impulsive setting and a Keplerian relative motion. We combine theoretical tools in optimal control with computer algebra and validated numerics: this brings guarantees on numerical solutions, while preserving efficiency.

In particular, we focus on the a posteriori validation of trajectories solutions of the linear differential equations of the dynamics. These are computed as truncated Chebyshev series together with rigorously computed error bounds. A theoretical and practical complexity analysis of an a posteriori quasi-Newton validation method is presented. Several representative examples show the advantages of our algorithms as well as their theoretical and practical limits.

This talk is based on joint works with D. Arzelier, F. Bréhard, N. Brisebarre, N. Deak, J.-B. Lasserre, C. Louembet, A. Rondepierre, B. Salvy, R. Serra.

July 15, 15:30 ~ 16:00 - Room 111

## Validation of KAM tori using a-posteriori formulation

### Alejandro Luque

### Instituto de Ciencias Matemáticas, Spain - luque@icmat.es

In this talk we present a methodology to rigorously validate a given approximation of a quasi-periodic Lagrangia torus of an exact symplectic map. That is, we check the hypothesis of an a-posteriori KAM theorem and we prove the existence of a true invariant torus nearby. Our method is sustained in the a-posteriori KAM formulation developed in the last decade by R. de la Llave and collaborators.

To check the hypotheses of the theorem, we use rigorous fast Fourier transform in combination with a sharp control of the discretization error. An important consequence is that the rigorous computations are performed in a very fast way. Indeed, with the same asymptotic cost of using the parameterization method to obtain numerical approximations of invariant tori.

We will discuss the application of the method to the standard map and the Froeschlé maps.

Joint work with J.-Ll. Figueras, A. Haro.

July 15, 16:00 ~ 16:30 - Room 111

## Model rejection and parameter reduction via time series

### Tomas Gedeon

### Montana State University, United States - tgedeon@gmail.com

We discuss a new approach to Dynamic Signatures Generated by Regulatory Networks (DSGRN) provides a queryable description of global dynamics over the entire parameter space. We perform a model validation within this class of dynamical systems. We show how a graph algorithm for finding matching labeled paths in pairs of labeled directed graphs can be used to reject models that do not match experimental time series. In particular, we extract a partial order of events describing local minima and local maxima of observed quantities from experimental time-series data from which we produce a labeled directed graph we call the pattern graph for which every path from root to leaf corresponds to a plausible sequence of events. We then consider the regulatory network model, which can be itself rendered into a labeled directed graph we call the search graph via techniques previously developed in computational dynamics. Labels on the pattern graph correspond to experimentally observed events, while labels on the search graph correspond to mathematical facts about the model. We give a theoretical guarantee that failing to find a match invalidates the model. As an application we consider gene regulatory models for the yeast S. cerevisiae.

Joint work with Bree Cummins (Montana State University), Shaun Harker (Rutgers University) and Konstantin Mischaikow (Rutgers University).

July 15, 17:00 ~ 17:30 - Room 111

## On computation of monodromy of the Hénon map and its applications

### Zin Arai

### Chubu University, Japan - zin@isc.chubu.ac.jp

We develop a computational method for the investigation of monodromy actions associated to loops in the parameter space of the complex Hénon map. When the loop is contained in the hyperbolic parameter region, we can determine the action rigorously as an automorphism of the shift space. The monodromy action is closely related to the structure of bifurcation curves in the parameter space and thus our method can be used to understand the bifurcations of the Julia sets.

July 15, 17:30 ~ 18:00 - Room 111

## The geometry of blenders in a three-dimensional Hénon-like family

### Stefanie Hittmeyer

### University of Auckland, New Zealand - stefanie.hittmeyer@auckland.ac.nz

Blenders are a geometric tool to construct complicated dynamics in diffeomorphisms of dimension at least three and vector fields of dimension at least four. They admit invariant manifolds that behave like geometric objects which have dimensions higher than expected from the manifolds themselves. We consider an explicit family of three-dimensional Hénon-like maps that exhibit blenders in a specific regime in parameter space. Using advanced numerical techniques we compute stable and unstable manifolds in this system, enabling us to show one of the first numerical pictures of the geometry of blenders. We furthermore present numerical evidence suggesting that the regime of existence of the blenders extends to a larger region in parameter space.

Joint work with Bernd Krauskopf (University of Auckland, New Zealand), Hinke Osinga (University of Auckland, New Zealand) and Katsutoshi Shinohara (Hitotsubashi University, Japan).

## Posters

## Approximation of basin of attraction in piecewise-affine systems by star-shaped polyhedral sets

### Maxim Demenkov

### Institute of Control Sciences, Russia - max.demenkov@gmail.com

We consider piecewise-affine system \[ \dot x(t)=A_ix(t) +f_i\;\mbox{ if }\; x(t)\in {\bf S}_i, \] and suppose that regions ${\bf S}_i, i=\overline{0,M}$ are polyhedral (i.e. defined by systems of linear inequalities) and the system state derivatives $\dot x(t)\in R^n$ are continuous on the boundaries between regions. We assume that $x_0 = 0$ is the stable equilibrium. In this case, an important system characteristic is the size of its basin of attraction for $x_0$.

We consider guaranteed estimation of the basin through the level sets of a Lyapunov function [1] - a positive definite continuous function $V(x)$, which is decreasing along the solution trajectories of the system. An estimation of the basin is given by a compact set defined as \[ {\bf \Omega}_\gamma=\{x | V(x)\leq \gamma\}. \] We employ star-shaped polyhedral functions $V(x)$ defined over union of convex cones ${\bf C}_j, j=\overline{1,N}$, having $x_0$ as a common point, so that in each cone the function is linear: \[ V(x)=d_j^Tx \;\mbox{ if }\; x\in {\bf C}_j, d_j\in R^n. \] The estimation of maximum size in terms of inclusion is given by the following choice of $\gamma$: \[ \gamma=\min_{x\in {\bf H}} V(x),\; {\bf H}=\{ x :D^{+}V(x(t))=0, x\neq 0\}, \] where $D^{+}V(x(t))$ is the upper Dini derivative of Lyapunov function [2] along system trajectories and ${\bf H}$ appears to be piecewise-affine manifold. The manifold ${\bf H}$ divides the state space into two (possibly disconnected) regions. We suppose that $x_0$ is in the region where $D^{+}V(x(t))<0$, so to find maximum estimation we are looking for scaling factor $\gamma$ so that the level set ${\bf \Omega}_\gamma$ fits in the region and touches the manifold ${\bf H}$. The notion of Dini derivative is actually a replacement of usual derivative of smooth Lyapunov function along trajectories of smooth systems [3].

[1] Chesi G., Domain of attraction. Analysis and control via SOS programming, Springer-Verlag, London (2011). [2] Giesl P., Hafstein S., Review on computational methods for Lyapunov functions, Discrete and continuous dynamical systems. Series B, 20, No.v8, 2291-2331 (2015). [3] Blanchini F., Miani S., Set-theoretic methods in control, Birkhauser Boston (2008).

## Numerical Continuation Study of Invariant Solutions of the Complex Ginzburg-Landau Equation

### Vanessa López

### IBM T. J. Watson Research Center, Yorktown Heights, NY, USA - lopezva@us.ibm.com

We consider the problem of computation and deformation of group orbits of solutions of the complex Ginzburg-Landau equation (CGLE) with cubic nonlinearity in $1\!+\!1$ space-time dimension invariant under the action of the three-dimensional Lie group of symmetries $A(x,t) \rightarrow \mathrm{e}^{\mathrm{i}\theta}A(x+\sigma,t+\tau)$. From an initial set of group orbits of such invariant solutions, for a particular point in the parameter space of the CGLE, we obtain new sets of group orbits of invariant solutions via numerical continuation along paths in the moduli space. The computed solutions along the continuation paths are unstable, and have multiple modes and frequencies active in their spatial and temporal spectra, respectively. Structural changes in the moduli space resulting in symmetry gaining / breaking associated mainly with the spatial reflection symmetry $A(x,t) \rightarrow A(-x,t)$ of the CGLE were frequently uncovered in the parameter regions traversed.

## Halo orbits and their bifurcations - rigorous numerical approach

### Irmina Walawska

### Jagiellonian University, Poland - irmina.walawska@uj.edu.pl

The Circular Restricted Three Body Problem (CR3BP) is a model equation for motion of a massless particle in the gravitational force of two large primaries. The system is given by $ \begin{cases} \ddot{x} - 2\dot y = \frac{\partial \Omega(x,y,z)}{\partial x},\\ \ddot{y} + 2\dot x = \frac{\partial \Omega(x,y,z)}{\partial y},\\ \ddot{z} = \frac{\partial \Omega(x,y,z)}{\partial z}, \end{cases} $ where $ \Omega(x,y,z) = \frac{1}{2}(x^2+y^2) + \frac{1-\mu}{d_1} + \frac{\mu}{d_2} $ and $ d_1=\sqrt{(x+\mu)^2+y^2+z^2},\\ d_2=\sqrt{(x-1+\mu)^2+y^2+z^2}, $ where $\mu$ denotes the relative mass ratio of the two main bodies. It was observed by Robert Farquhar that there is a family of symmetric, periodic orbits, parameterized by the amplitude $z$. These orbits are called Halo orbits. We propose an algorithm for rigorous validation of bifurcation of family of periodic orbits. We also give an algorithm for rigorous continuation of these orbits. The method uses rigorous computation of higher order derivatives of well chosen Poincar\'e map with symmetry properties of the system. As an application we give a computer assisted proof that the Halo orbits bifurcate from the family of Lyapunov orbits for wide range of the parameters $\mu$. For $\mu$ corresponding to the Sun-Jupiter system we give a proof of the existence of a wide continuous branch of Halo orbits that undergo period doubling, trippling and quadrluping bifurcation for some amplitude $z$. The computer assisted proof uses rigorous ODE solvers and algorithms for computation of Poincare maps from the CAPD library.

Joint work with Daniel Wilczak, Jagiellonian University, Poland.

## Spectral methods for transfer operators in one-dimensional dynamics

### John P Wormell

### The University of Sydney, Australia - j.wormell@maths.usyd.edu.au

Chaotic maps of the interval are important models in dynamical systems, and quantitative aspects of their long-time statistical properties (such as invariant measures) are key objects of study. These properties can be found by solving linear problems involving the transfer operator, but existing numerical methods converge very slowly and cannot capture many important statistical properties.

We present a rigorously justified Chebyshev spectral method for calculating statistical properties in the case of Markovian uniformly-expanding maps. This spectral method can be neatly integrated into an infinite-dimensional linear algebra framework. As a result, the accurate calculation of any statistical property of interest is not only quick and reliable, but highly convenient.

The spectral method can also be extended through inducing schemes to many other kinds of maps of the interval. We present the specific case of intermittent maps, which, though of much interest theoretically, have been hitherto been largely out of reach of numerical study.