Related papers: Existence of Chaos in Evolution Equations

This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are…

We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=\mu>0$ shadowing chains of homoclinic orbits to…

In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also…

Let a system of differential equations possess a saddle-node periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation…

The cyclic evolutions and associated geometric phases induced by time-independent Hamiltonians are studied for the case when the evolution operator becomes the identity (those processes are called {\it evolution loops}). We make a detailed…

We examine a 2DOF Hamiltonian system, which arises in study of first-order mean motion resonance in spatial circular restricted three-body problem "star-planet-asteroid", and point out some mechanisms of chaos generation. Phase variables of…

Planetary, stellar and galactic physics often rely on the general restricted gravitational N-body problem to model the motion of a small-mass object under the influence of much more massive objects. Here, I formulate the general restricted…

A mathematical model is constructed for the evolution of spherical perturbations in a cosmological one-component statistical system of completely degenerate scalarly charged fermions with a scalar Higgs interaction. A complete system of…

We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an…

We explore the quantum scattering of systems classically described by binary and other low order Smale horseshoes, in a stage of development where the stable island associated with the inner periodic orbit is large, but chaos around this…

We consider the planar restricted $N$-body problem where the $N-1$ primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal…

For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can poten- tially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it…

In this paper we study a type of two singular point singular cycle where one heteroclinic orbit is the transversal intersection of the 2-dimensional stable manifold of one singular point and the 2-dimensional unstable manifold of other…

In this paper, we develop a way of analyzing the random dynamics of stochastic evolution equations with a non-dense domain. Such problems cover several types of evolution equations. We are particularly interested in evolution equations with…

For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise,…

We consider infinite-dimensional parabolic rough evolution equations. Using regularizing properties of analytic semigroups we prove global-in-time existence of solutions and investigate random dynamical systems for such equations.

We investigate species-rich mathematical models of ecosystems. While much of the existing literature focuses on the properties of equilibrium fixed points, persistent dynamics (e.g., limit cycles or chaos) have also been observed, both in…

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…

In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the…

Semilinear stochastic evolution equations with multiplicative L\'evy noise and monotone nonlinear drift are considered. Unlike other similar work we do not impose coercivity conditions on coefficients. Existence and uniqueness of the mild…