The classical Lyapunov-Poincaré center theorem assures the existence of a first integral for an analytic one-form near a center singularity in dimension two, provided that the first jet of the one-form is nondegenerate. The basic point is the existence of an analytic first integral for the given one-form. In this paper we consider generalizations for two main frameworks: (i) real analytic foliations of codimension one in higher dimension and (ii) singular holomorphic foliations in dimension two. All this is related to the problem of finding criteria assuring the existence of analytic first integrals for a given codimension one germ with a suitable first jet. Our approach consists in giving an interpretation of the center theorem in terms of holomorphic foliations and, following an idea of Moussu, apply the holomorphic foliations arsenal in the obtaining the required first integral. As a consequence we are able to revisit some of Reeb classical results on integrable perturbations of exact homogeneous one-forms, and prove some versions of these to the framework of non-isolated (perturbations of transversely Morse type) singularities.
Let \Fc be a holomorphic foliation by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: (1) the singular points of \Fc are all hyperbolic; (2) \Fc is Brody hyperbolic. Then we establish cohomological formulas for the Lyapunov exponent and the Poincaré mass of an extremal positive \ddc-closed current tangent to \Fc. If, moreover, there is no nonzero positive closed current tangent to \Fc, then we show that the Lyapunov exponent χ(\Fc) of \Fc, which is, by definition, the Lyapunov exponent of the unique normalized positive \ddc-closed cur…
We study chaotic dynamics in a system of four differential equations describing the dynamics of five identical globally coupled phase oscillators with biharmonic coupling. We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of the three-dimensional Poincaré map for the system under consideration. We show that the chaotic dynamics develop here near a codimension three bifurcation, when a per…
One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations (\cite{moussu}). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in di…
Using Galois theory of functional equations, we give a new proof of the main result of the paper "Transcendental transcendency of certain functions of Poincaré" by J.F. Ritt, on the differential transcendence of the solutions of the functional equation R(y(t))=y(qt), where R is a rational function with complex coefficients which verifies R(0)=0, R'(0)=q, where q is a complex number with |q|>1. We also give a partial result in the case of an algebraic function R.
We investigate the singularly perturbed monotone systems with respect to cones of rank $2$ and obtain the so called Generic Poincaré-Bendixson theorem for such perturbed systems, that is, for a bounded positively invariant set, there exists an open and dense subset $\mathcal{P}$ such that for each $z\in \mathcal{P}$, the $ω$-limit set $ω(z)$ that contains no equilibrium points is a closed orbit.
Our first purpose is to study the stability of linear flows on real, connected, compact, semisimple Lie groups. After, we study and classify periodic orbits of linear and invariant flows. In particular, we obtain a version of Poincaré-Bendixon's Theorem. As an application, we present periodic orbits of linear or invariant flows on $SO(3)$ or $SU(2)$, and we classify periodic orbits of a linear or invariant system on $SO(4)$.
Covariant Lyapunov vectors (CLVs) are intrinsic modes that describe long-term linear perturbations of solutions of dynamical systems. With recent advances in the context of semi-invertible multiplicative ergodic theorems, existence of CLVs has been proved for various infinite-dimensional scenarios. Possible applications include the derivation of coherent structures via transfer operators or the stability analysis of linear perturbations in models of increasingly higher resolutions. We generalize the concept of Ginelli's algorithm to compute CLVs in Hilbert spaces. Our main result is a converge…
Forward invariance of a basin of attraction is often overlooked when using a Lyapunov stability theorem to prove local stability; even if the Lyapunov function decreases monotonically in a neighborhood of an equilibrium, the dynamic may escape from this neighborhood. In this note, we fix this gap by finding a smaller neighborhood that is forward invariant. This helps us to prove local stability more naturally without tracking each solution path. Similarly, we prove a transitivity theorem about basins of attractions without requiring forward invariance. Keywords: Lyapunov function, local stabil…
It is shown that the Poincaré-Birkhoff fixed point theorem may be proven by extending the geometric approach originally devised by Henri Poincaré himself, along with several results from elementary differential topology. Beginning with an example application of the theorem, we proceed by systematically constructing and classifying a certain set of invariant curves and their critical points. This classification is then used to prove the correctness of a procedure which guarantees the existence of at least two fixed points for any twist map of the annulus admitting a positive integral invariant.
Poincaré recurrence theorem implies the density of recurrent points for volume-preserving dynamical systems on compact domains. The density of closed orbits in the non-wandering set is one of the essential properties of Axiom A and chaos. The minimal flows are one of the most fundamental objects in topological dynamics. To analyze minimality, recurrence, and density in this paper, we introduce a strict limit circuit and a circuit with wandering holonomy. Using these concepts, we classify non-recurrent non-wandering orbits of flows on compact surfaces with finitely many singular points, charact…
We prove a generalization of the classical Poincaré--Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem (as proved by the authors in arXiv:2011.10386).
A rigorous proof of a theorem on the coexistence of smooth Lyapunov function and smooth planar dynamical system with one arbitrary limit cycle is given, combining with a novel decomposition of the dynamical system from the perspective of mechanics. We base on this dynamic structure incorporating several efforts of this dynamic structure on fixed points, limit cycles and chaos, as well as on relevant known results, such as Schoenflies theorem, Riemann mapping theorem, boundary correspondence theorem and differential geometry theory, to prove this coexistence. We divide our procedure into three …
Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincaré Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Cesàro mean associated with the iterates of the process. Applied to processes in $L^1(Ω,{\mathcal A},μ)$, where $μ$ is a probability measure, new conditional versions…
Given an i.i.d. sequence $\{A_n(ω)\}_{n\ge 1}$ of invertible matrices and a random matrix $B(ω)$, we consider the random matrix sequences inductively defined by $S_n(ω) = A_n(ω)S_{n-1}(ω)$ and $T_n(ω) = B(σ^{n-1}ω)S_{n-1}(ω)+A_n(ω)T_{n-1}(ω)$, and study several limit theorems involving $T_n(ω)$ as well as the asymptotic behaviour of the action of $T_n(ω)$ on the projective space and on the unit circle.
A degenerate dynamical system is characterized by a state-dependent multiplier of the time derivative of the state in the time evolution equation. It can give rise to Hamiltonian systems whose symplectic structure possesses a non-constant rank throughout the phase space. Around points where the multiplier becomes singular, flow can experience abrupt and irreversible changes. We introduce a topological index for degenerate dynamical systems around these {\it degeneracy points} and show that it refines and extends the usual topological index in accordance with the Poincaré-Hopf Theorem.
Optimal Initialization in Depth: Lyapunov Initialization and Limit Theorems for Deep Leaky ReLU Networks
School of Mathematics, Institute for Advanced Study（数学系，高级研究院）；Mathematical Institute, University of Oxford（牛津大学数学学院）；Max Planck Institute for Multidisciplinary Sciences（多学科科学研究所）；Department of Mathematics, University College London（伦敦大学学院数学系）
