Related papers: Averaging theory and catastrophes
Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in…
The stability of dynamical systems with oscillatory behaviors and well-defined average vector fields has traditionally been studied using averaging theory. These tools have also been applied to hybrid dynamical systems, which combine…
We review recent developments in structural stability as applied to key topics in general relativity. For a nonlinear dynamical system arising from the Einstein equations by a symmetry reduction, bifurcation theory fully characterizes the…
We discuss tipping phenomena (critical transitions) in nonautonomous systems using an example of a bistable ecosystem model with environmental changes represented by time-varying parameters [Scheffer et al., Ecosystems, 11 (2008), pp.…
This work proposes a data-driven approach for bifurcation analysis in nonlinear systems when the governing differential equations are not available. Specifically, regularized regression with barrier terms is used to learn a homeomorphism…
We describe the resulting spatiotemporal dynamics when a homogeneous equilibrium loses stability in a spatially extended system. More precisely, we consider reaction-diffusion systems, assuming only that the reaction kinetics undergo a…
The Birkhoff Ergodic Theorem asserts under mild conditions that Birkhoff averages (i.e. time averages computed along a trajectory) converge to the space average. For sufficiently smooth systems, our small modification of numerical Birkhoff…
We consider a nonlinear pendulum whose suspension point undergoes stochastic vibrations in its plane of motion. Stochastic vibrations are constructed by stochastic differential equations with random periodic solutions. Averaging over these…
The well-defined but intricate course of time evolution exhibited by many naturally occurring phenomena suggests some source of dynamic order sustaining it. In spite of its obviousness as a problem, it has remained absent from the…
Decision making is a fundamental capability of autonomous systems. As decision making is a process which happens over time, it can be well modeled by dynamical systems. Often, decisions are made on the basis of perceived values of the…
Important information about the dynamical structure of a differential system can be revealed by looking into its invariant compact manifolds, such as equilibria, periodic orbits, and invariant tori. This knowledge is significantly increased…
In this work, we obtain an a-posteriori theorem for the existence of partly hyperbolic invariant tori in analytic Hamiltonian systems: autonomous, periodic, and quasi-periodic. The method of proof is based on the convergence of a KAM…
In the first part of this paper, we consider a family of continuous-time dynamical systems coupled with diffusion-transmutation processes. Under certain conditions, such randomly perturbed dynamical systems can be interpreted as an averaged…
A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes not related to…
We present stability and recurrence results for a class of stochastic hybrid dynamical systems with oscillating flow maps. These results are developed by introducing averaging tools that parallel those already existing for ordinary…
Bifurcations of dynamical systems, described by a second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to…
We combine results available in the literature to prove that the torus emerging in a secondary Hopf bifurcation is normally hyperbolic. This result is then applied to establish sufficient conditions for the bifurcation of normally…
We study dynamics in a neighborhood of a nonhyperbolic fixed point or an irreducible homoclinic tangent point. General type conditions for the existence of infinite sets of periodic points are obtained. A new method, based on the study of…
In a nonautonomous nonlinear dynamical system, generic critical transitions (tipping points) are not limited to slow passage through fold bifurcations. They can also correspond to slow passage through other generic bifurcations, such as…
We present a geometric proof of the averaging theorem for perturbed dynamical systems on a Riemannian manifold, in the case where the flow of the unperturbed vector field is periodic and the $\mathbb{S}^{1}$-action associated to this vector…