The Lyapunov exponents and the neighbourhood of periodic orbits

ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to show that stable, simply unstable, and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially, and fully chaotic orbits in their neighbourhood.

Bifurcation Dynamics and Pattern Formation of a Three-Species Food Chain System with Discrete Spatiotemporal Variables

The bifurcation dynamics and pattern formation of a discrete-time three-species food chain system with Beddington–DeAngelis functional response are investigated. Via applying the centre manifold theorem and bifurcation theorems, the occurrence conditions for flip bifurcation and Neimark–Sacker bifurcation as well as Turing instability are determined. Numerical simulations verify the theoretical results and reveal many interesting dynamic behaviours. The flip bifurcation and the Neimark–Sacker bifurcation both induce routes to chaos, on which we find period-doubling cascades, invariant curves, chaotic attractors, sub–Neimark–Sacker bifurcation, sub–flip bifurcation, chaotic interior crisis, sub–period-doubling cascade, periodic windows, sub–periodic windows, and various periodic behaviours. Moreover, the food chain system exhibits various self-organized patterns, including regular and irregular patterns of stripes, labyrinth, and spiral waves, suggesting the populations can coexist in space as many spatiotemporal structures. These analysis and results provide a new perspective into the complex dynamics of discrete food chain systems.

Local Stability in 3D Discrete Dynamical Systems: Application to a Ricker Competition Model

A survey on the conditions of local stability of fixed points of three-dimensional discrete dynamical systems or difference equations is provided. In particular, the techniques for studying the stability of nonhyperbolic fixed points via the centre manifold theorem are presented. A nonlinear model in population dynamics is studied, namely, the Ricker competition model of three species. In addition, a conjecture about the global stability of the nontrivial fixed points of the Ricker competition model is presented.

BIFURCATION ANALYSIS OF A LOGISTIC PREDATOR–PREY SYSTEM WITH DELAY

We consider a coupled, logistic predator–prey system with delay. Mainly, by choosing the delay time${\it\tau}$as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time${\it\tau}$passes some critical values. Based on the normal-form theory and the centre manifold theorem, we also derive formulae to obtain the direction, stability and the period of the bifurcating periodic solution at critical values of ${\it\tau}$. Finally, numerical simulations are investigated to support our theoretical results.

HOPF BIFURCATION ANALYSIS FOR A RATIO-DEPENDENT PREDATOR–PREY SYSTEM INVOLVING TWO DELAYS

The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

Hopf Bifurcation and Numerical Simulation in a Calcium Oscillation Model

Calcium oscillations play a very important role in providing the intracellular signaling, and many mathematical models have been proposed to describe calcium oscillations. The Shen-Larter model presented here is based on calcium-induced calcium release (CICR) and the inositol trisphosphate cross-coupling (ICC). Nonlinear dynamics of this model is investigated by using the centre manifold theorem and bifurcation theory, including the variation in classification and stability of equilibria with different parameter values. The results show that the appearance and disappearance of calcium oscillations are due to subcritical Hopf bifurcation of equilibria. The numerical simulations are performed in order to illustrate the correctness of our theoretical analysis, including the bifurcation diagram of fixed points, the phase diagram of the system in two dimensional space and time series.

Analysis of Degenerate Bifurcation in Machining Using a Nonlinear Model

We present a two degrees of freedom model of machining dynamics that compactly captures the nonlinear effects of regeneration, tool wear and plowing under orthogonal machining. Extensive simulation experiments show that the solutions to the governing equations of the model bear distinct similarities to signals from our earlier experiments, as reflected by both visual state portraits as well as the values of the quantifiers of a steady state dynamic system. Governing equations of the model lead to nonlinear delay differential equations, which we reduce to ordinary differential equations using Hopf bifurcation theorem and centre manifold theorem. Despite the ongoing efforts by the authors of this paper to quantify the simulations results analytically and experimentally, we strongly believe that our proposed model will be found to be amenable for studying and analyzing bifurcations that can lead to chatter in machining.

A centre manifold theorem for hyperbolic PDEs

SynopsisA version of the centre manifold theorem is established which is suitable for quasilinear hyperbolic equations. As an application, the Benard problem for a viscoelastic fluid is discussed.

Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow

The stability conditions of plane waves against three-dimensional perturbations in plane Poiseuille flow, as described by a dispersive cubically nonlinear complex-amplitude equation, under perturbations quasi-periodic in two of the space dimensions are investigated. It is found that if the parameters satisfy certain conditions, a wave is totally stable. These conditions are an extension of those given for the lower dimensional case by J. T. Stuart and R. C. DiPrima ( Proc R. Soc. Lond . A 362, 27-41 (1978)). The centre manifold theorem is then used to investigate the nature of the solutions bifurcating from a marginally unstable plane wave. Hopf bifurcations occur in the 1, 2 or 3 perturbing sidebands that are neutrally stable to the unperturbed wave and can give rise to limit cycles or tori.