### Workshop B1 - Computational Dynamics

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

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

### 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).

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

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

### 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).

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

### 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).

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

### 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.

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## Halo orbits and their bifurcations - rigorous numerical approach

### 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.

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

### 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.

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FoCM 2017, based on a nodethirtythree design.