Unstable Manifolds Of Saddle Research Articles

A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

We address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.

Analytical estimates of the effect of amplitude modulated signal in nonlinearly damped Duffing-vander Pol oscillator

The effect of amplitude modulated (AM) signal on horseshoe chaos is studied both analytically and numerically in nonlinearly damped Duffing-vander Pol (DVP) oscillator. We assume the nonlinear damping term is proportional to the power of velocity (x˙) in the form |x˙|p−1. Using the Melnikov analytical method, the analytical threshold condition for the prediction of onset of horseshoe chaos is obtained. Melnikov threshold curves are drawn in different external parameters space. Our results show that as the damping exponent p increases from a small value, the threshold value for the onset of horseshoe is decreased. The analytical results are verified by numerical simulations using the computation of stable and unstable manifolds of saddle, measuring the time elapsed between two successive transverse intersections, bifurcation diagram, maximal Lyapunov exponent, phase portrait and Poincare´ map.

Chaos prediction and control of Goodwin’s nonlinear accelerator model

Chaos prediction and its control of the Goodwin model under the deterministic or stochastic excitation are studied theoretically and numerically. Applying the Melnikov technique, the threshold conditions for the occurrence of chaos are obtained theoretically. The stable and unstable manifolds of saddle are computed to verify the effectiveness of the analytical prediction in the deterministic case. Also, the safe basins are introduced to show how the externally stochastic perturbation affects the safety of the economic system as the noise amplitude increases. Finally, the analytical criterion of controlling chaos is derived via the delayed feedback control method. Numerical investigations including the top Lyapunov exponent, Poincare section, and phase portraits are carried out to demonstrate the validity and effectiveness of the theoretical results.

Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude-modulated force

The homoclinic bifurcation and transition from regular to asymptotic chaos in Duffing oscillator subjected to an amplitude modulated force is studied both analytically and numerically. Applying the Melnikov analytical method, the threshold condition for the occurrence of horseshoe chaos is obtained. Melnikov threshold curves are drawn in different external parameters space. Analytical predictions are demonstrated through direct numerical simulations. Parametric regimes where suppression of horseshoe chaos occurs are predicted. Period doubling route to chaos, intermittency route to chaos and quasiperiodic route to chaos are found to occur due to the amplitude-modulated force. Numerical investigations including computation of stable and unstable manifolds of saddle, maximal Lyapunov exponent, Poincaré map and bifurcation diagrams are used to detect homoclinic bifurcation and chaos.

Structure of the attracting set of a piecewise linear Hénon mapping

Detailed structure of the attracting set of the piecewise linear Henon mapping (x,y)→(1−a|x|+by,x) with a=8/5 and b=9/25 is described in this paper using the method of dual line mapping. Let A and B denote the fixed saddles in the first quadrant, and in the third quadrant, respectively. It is claimed that (1) the attracting set is the closure of the unstable manifold of saddle B, which includes the unstable manifold of A as its subset, and (2) the basin of attraction is the closure of the stable manifold of A, bounded by the stable manifold of B, which is in the limiting set of the stable manifold of A. Relations of the manifolds of the periodic saddles with the manifolds of the fixed point are given. Symbolic dynamics notations are adopted which renders possible the study of the dynamical behavior of every piece of the manifolds and of every homoclinic or heteroclinic point.