Li-Yorke chaos of linear differential equations in a finite-dimensional space with a weak topology.

Li-Yorke chaos of linear differential equations in a finite-dimensional space with a weak topology is introduced. Based on this topology on the Euclidean space, a flow generated from a linear differential equation is proved to be Li-Yorke chaotic under certain conditions, which is in sharp contract to the well-known fact that linear differential equations cannot be chaotic in a finite-dimensional space with a strong topology.

Nonisotropic chaotic vibrations of a 2D hyperbolic PDE.

Little seems to be known about the chaos of the two-dimensional (2D) hyperbolic partial differential equations (PDEs). The objective of this paper is to study the nonisotropic chaotic vibrations of a system governed by a 2D linear hyperbolic PDE with mixed derivative terms (MDTs) and a nonlinear boundary condition (NBC), where the interaction between MDTs and NBC causes the energy of such a system to rise and fall. The 2D hyperbolic system is proved to be topologically conjugate with the corresponding Riemann invariants, which are rigorously proved to be chaotic. Two numerical examples are carried out to demonstrate the theoretical results.

Singular cycles and chaos in a new class of 3D three-zone piecewise affine systems.

It is a great challenge to detect singular cycles and chaos in dynamical systems with multiple discontinuous boundaries. This paper takes the challenge to investigate the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles connecting saddle-focus equilibriums, in a new class of three-dimensional three-zone piecewise affine systems. It develops a method to accurately predict the coexisting homoclinic and heteroclinic cycles in such a system. Furthermore, this paper establishes some conditions for chaos to exist in the system, with rigorous mathematical proof of chaos emerged from the coexistence of these singular cycles. Finally, it presents numerical simulations to verify the theoretical results.

Bifurcation dynamics of the tempered fractional Langevin equation.

Tempered fractional processes offer a useful extension for turbulence to include low frequencies. In this paper, we investigate the stochastic phenomenological bifurcation, or stochastic P-bifurcation, of the Langevin equation perturbed by tempered fractional Brownian motion. However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework. We first derive the spectral density function of the considered system based on the generalized Parseval's formula and the Wiener-Khinchin theorem. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosive-like, unimodal, and bimodal kurtosis. Therefore, our procedures in this paper are not merely comparable in scope to the existing theory of Markovian systems but also provide a possible approach to discern P-bifurcation dynamics in the non-Markovian settings.

Dynamics of the stochastic Lorenz chaotic system with long memory effects.

Little seems to be known about the ergodic dynamics of stochastic systems with fractional noise. This paper is devoted to discern such long time dynamics through the stochastic Lorenz chaotic system (SLCS) with long memory effects. By a truncation technique, the SLCS is proved to generate a continuous stochastic dynamical system Λ. Based on the Krylov-Bogoliubov criterion, the required Lyapunov function is further established to ensure the existence of the invariant measure of Λ. Meanwhile, the uniqueness of the invariant measure of Λ is proved by examining the strong Feller property, together with an irreducibility argument. Therefore, the SLCS has exactly one adapted stationary solution.

Variational solutions and random dynamical systems to SPDEs perturbed by fractional Gaussian noise.

This paper deals with the following type of stochastic partial differential equations (SPDEs) perturbed by an infinite dimensional fractional Brownian motion with a suitable volatility coefficient Φ: dX(t) = A(X(t))dt+Φ(t)dB (H) (t), where A is a nonlinear operator satisfying some monotonicity conditions. Using the variational approach, we prove the existence and uniqueness of variational solutions to such system. Moreover, we prove that this variational solution generates a random dynamical system. The main results are applied to a general type of nonlinear SPDEs and the stochastic generalized p-Laplacian equation.

Fractional noise destroys or induces a stochastic bifurcation.

Little seems to be known about the stochastic bifurcation phenomena of non-Markovian systems. Our intention in this paper is to understand such complex dynamics by a simple system, namely, the Black-Scholes model driven by a mixed fractional Brownian motion. The most interesting finding is that the multiplicative fractional noise not only destroys but also induces a stochastic bifurcation under some suitable conditions. So it opens a possible way to explore the theory of stochastic bifurcation in the non-Markovian framework.

Periodic solutions and bifurcation in an S I S epidemic model with birth pulses.

The dynamical behavior of an S I S epidemic model with birth pulses and a varying population is discussed analytically and numerically. This paper investigates the existence and stability of the infection-free periodic solution and the endemic periodic solution. By using discrete maps, the center manifold theorem, and the bifurcation theorem, the conditions of existence for bifurcation of the positive periodic solution are derived. Moreover, numerical results for phase portraits, periodic solutions, and bifurcation diagrams, which are illustrated with an example, are in good agreement with the theoretical analysis.

A simple time-delay feedback anticontrol method made rigorous.

An effective method of chaotification via time-delay feedback for a simple finite-dimensional continuous-time autonomous system is made rigorous in this paper. Some mathematical conditions are derived under which a nonchaotic system can be controlled to become chaotic, where the chaos so generated is in a rigorous mathematical sense of Li-Yorke in terms of the Marotto theorem. Numerical simulations are given to verify the theoretical analysis.