Star flows with singularities of different indices

It is known that a generic star vector field $X$ on a $3$ or $4$-dimensional manifold is such that its chain recurrence classes are either hyperbolic, or singular hyperbolic ([MPP] and [GSW]). Palis conjectured that every vector field must be approximated either by singular hyperbolic vector fields or by vector fields with homoclinic tangencies or heterodimensional cycles (associated to periodic orbits). We give a counter example in dimension $5$ (and higher). We present here an open set of star vector fields on a $5$-dimensional manifold for which two singular points with different indices belong (robustly) to the same chain recurrence class. This prevents the class to be singular hyperbolic, showing that the results in [MPP] can not be extended to higher dimensions and thus contradicting the conjecture by Palis.Comment: The paper was rearranged and overhaule

Geometric Analysis of the Formation Problem for Autonomous Robots

In the formation control problem for autonomous robots a distributed control law steers the robots to the desired target formation. A local stability result of the target formation can be derived by methods of linearization and center manifold theory or via a Lyapunov-based approach. It is well known that there are various other undesired invariant sets of the robots' closed-loop dynamics. This paper addresses a global stability analysis by a differential geometric approach considering invariant manifolds and their local stability properties. The theoretical results are then applied to the well-known example of a cyclic triangular formation and result in instability of all invariant sets other than the target formation.Comment: submitted to the IEEE Transaction on Automatic Contro

Ergodic optimization of Birkhoff averages and Lyapunov exponents

Ergodic optimization is the study of extremal values of asymptotic dynamical quantities such as Birkhoff averages or Lyapunov exponents, and of the orbits or invariant measures that attain them. We discuss some results and problems.Comment: Paper for ICM 2018 Proceedings. 21 pages, 1 figure. 3rd version: A few corrections were mad

Global linearization and fiber bundle structure of invariant manifolds

We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict our attention to continuous-time dynamical systems, or flows. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a $C^k$ disk bundle structure if the local stable foliation is assumed $C^k$. We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global $C^k$ linearizing conjugacy. We also prove a $C^k$ global linearization result for inflowing NAIMs; we believe that even the local version of this result is new, and may be useful in applications to slow-fast systems. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.Comment: 40 pages, 4 figures. Version as accepted for publication with only minor change

Random Products of Standard Maps

We develop a general geometric method to establish the existence of positive Lyapunov exponents for a class of skew products. The technique is applied to show non-uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps, notably for skew-products whose fiber dynamics is given by (a continuum of parameters in) the Froeschl\'e family. These types of coupled systems appear as some induced maps in models for the study of Arnold diffusion. Consequently, we are able to present new examples of partially hyperbolic diffeomorphisms having rich high dimensional center dynamics. The methods are also suitable for studying cocycles over shift spaces, and do not demand any low dimensionality condition on the fiber.Comment: References updated. 33 pages. To be published in CM

The geometric approach for constructing Sinai-Ruelle-Bowen measures

An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric `push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems.Comment: 30 page

A Framework for Planning and Controlling Non-Periodic Bipedal Locomotion

This study presents a theoretical framework for planning and controlling agile bipedal locomotion based on robustly tracking a set of non-periodic apex states. Based on the prismatic inverted pendulum model, we formulate a hybrid phase-space planning and control framework which includes the following key components: (1) a step transition solver that enables dynamically tracking non-periodic apex or keyframe states over various types of terrains, (2) a robust hybrid automaton to effectively formulate planning and control algorithms, (3) a phase-space metric to measure distance to the planned locomotion manifolds, and (4) a hybrid control method based on the previous distance metric to produce robust dynamic locomotion under external disturbances. Compared to other locomotion frameworks, we have a larger focus on non-periodic gait generation and robustness metrics to deal with disturbances. Such focus enables the proposed control framework to robustly track non-periodic apex states over various challenging terrains and under external disturbances as illustrated through several simulations. Additionally, it allows a bipedal robot to perform non-periodic bouncing maneuvers over disjointed terrains.Comment: 33 pages, 18 figures, journa

Exponential stability for nonautonomous functional differential equations with state-dependent delay

The properties of stability of compact set $\mathcal{K}$ which is positively invariant for a semiflow $(\Omega\times W^{1,\infty}([-r,0],\mathbb{R}^n),\Pi,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $\mathcal{K}$ induce linear skew-product semiflows on the bundles $\mathcal{K}\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and $\mathcal{K}\times C([-r,0],\mathbb{R}^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of $\mathcal{K}$ in $\Omega\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and also to the exponential stability of this minimal set when the supremum norm is taken in $W^{1,\infty}([-r,0],\mathbb{R}^n)$. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions

Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations

This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our approximation, we decompose the stable manifold into three components: a finite dimensional slow component, a fast-but-finite dimensional component, and a strongly contracting infinite dimensional "tail". We employ the parameterization method in a finite dimensional projection to approximate the slow-stable manifold, as well as the attached finite dimensional invariant vector bundles. This approximation provides a change of coordinates which largely removes the nonlinear terms in the slow stable directions. In this adapted coordinate system we apply the Lyapunov-Perron method, resulting in mathematically rigorous bounds on the approximation errors. As a result, we obtain significantly sharper bounds than would be obtained using only the linear approximation given by the eigendirections. As a concrete example we illustrate the technique for a 1D Swift-Hohenberg equation.Comment: 57 pages, 3 figure

KAM theory for some dissipative systems

Dissipative systems play a very important role in several physical models, most notably in Celestial Mechanics, where the dissipation drives the motion of natural and artificial satellites, leading them to migration of orbits, resonant states, etc. Hence the need to develop theories that ensure the existence of structures such as invariant tori or periodic orbits and device efficient computational methods. In this work we concentrate on the existence of invariant tori for the specific case of dissipative systems known as "conformally symplectic" systems, which have the property that they transform the symplectic form into a multiple of itself. To give explicit examples of conformally symplectic systems, we will present two different models: a discrete system, the standard map, and a continuous system, the spin-orbit problem. In both cases we will consider the conservative and dissipative versions, that will help to highlight the differences between the symplectic and conformally symplectic dynamics. For such dissipative systems we will present a KAM theorem in an a-posteriori format. The method of proof is based on extending geometric identities originally developed in [39] for the symplectic case. Besides leading to streamlined proofs of KAM theorem, this method provides a very efficient algorithm which has been implemented. Coupling an efficient numerical algorithm with an a-posteriori theorem, we have a very efficient way to provide rigorous estimates close to optimal. Indeed, the method gives a criterion (the Sobolev blow up criterion) that allows to compute numerically the breakdown. We will review this method as well as an extension of J. Greene's method and present the results in the conservative and dissipative standard maps. Computing close to the breakdown, allows to discover new mathematical phenomena such as the "bundle collapse mechanism".Comment: 40 pages, 10 figure