Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-fractional and their nonlinearities tend to cancel leaving only a small total, then they can all be replaced with affine maps with the same domains and images and the resulting composition is a very good approximation of the original one. These technical tools are then applied to prove a theorem about critical circle maps.
Poincaré maps and suspension flows are examples of fundamental constructions in the study of dynamical systems. This study aimed to show that these constructions define an adjoint pair of functors if categories of dynamical systems are suitably set. First, we consider the construction of Poincaré maps in the category of flows on topological manifolds, which are not necessarily smooth. We show that well-known results can be generalized and the construction of Poincaré maps is functorial, if a category of flows with global Poincaré sections is adequately defined. Next, we consider the constructi…
In this article we present recent advances on interval methods for rigorous computation of Poincaré maps. We also discuss the impact of choice of Poincaré section and coordinate system on obtained bounds for computing Poincaré map nearby fixed points.
Multi-shooting parameterization methods for invariant manifolds and heteroclinics of 2 DOF Hamiltonian Poincaré maps, with applications to celestial resonant dynamics
For a class of flows on polytopes, including many examples from Evolutionary Game Theory, we describe a piecewise linear model which encapsulates the asymptotic dynamics along the heteroclinic network formed out of the polytope's vertexes and edges. This piecewise linear flow is easy to compute even in higher dimensions, which allows the usage of numeric algorithms to find invariant dynamical structures such as periodic, homoclinic or heteroclinic orbits, which if robust persist as invariant dynamical structures of the original flow. We apply this method to prove the existence of chaotic behav…
Poincaré maps are an integral aspect to our understanding and analysis of nonlinear dynamical systems. Despite this fact, the construction of these maps remains elusive and is primarily left to simple motivating examples. In this manuscript we propose a method of data-driven discovery of Poincaré maps based upon sparse regression techniques, specifically the sparse identification of nonlinear dynamics (SINDy) algorithm. This work can be used to determine the dynamics on and near invariant manifolds of a given dynamical system, as well as provide long-time forecasting of the coarse-grained dyna…
A theorem on computation of the homological Conley index of an isolated invariant set of the Poincaré map associated to a section in a rotating local dynamical system $ϕ$ is proved. Let $(N,L)$ be an index pair for a discretization $ϕ^h$ of $ϕ$, where $h>0$, and let $S$ denote the invariant part of $N\setminus L$; it follows that the section $S_0$ of $S$ is an isolated invariant set of the Poincaré map. The theorem asserts that if the sections $N_0$ of $N$ and $L_0$ of $L$ are ANRs, the homology classes $[u_j]$ of some cycles $u_j$ form a basis of $H(N_0,L_0)$, and for some scalars $a_{ij}$, t…
We present a higher-dimensional version of the Poincaré-Birkhoff theorem which applies to Poincaré time maps of Hamiltonian systems. The maps under consideration are neither required to be close to the identity nor to have a monotone twist. The annulus is replaced by the product of an $N$-dimensional torus and the interior of a $(N-1)$-dimensional (not necessarily convex) embedded sphere; on the other hand, the classical boundary twist condition is replaced by an avoiding rays condition.
The intrinsic nature of a problem usually suggests a first suitable method to deal with it. Unfortunately, the apparent ease of application of these initial approaches may make their possible flaws seem to be inherent to the problem and often no alternative ways to solve it are searched for. For instance, since linear systems of differential equations are easy to integrate, Poincaré half-maps for piecewise linear systems are always studied by using the direct integration of the system in each zone of linearity. However, this approach is accompanied by two important defects: due to the differen…
In this paper we prove the existence of multiple periodic solutions (harmonic and subharmonic) for a class of planar Hamiltonian systems which include the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory of bend-twist maps and topological horseshoes. The case of complex dynamics is investigated, too.
