Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow

We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold [Formula: see text] are the open cells of a CW-decomposition of [Formula: see text].

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.

Two-Dimensional Manifolds of Modified Chen System with Time Delay

Two-dimensional unstable manifolds of the modified Chen system are constructed at equilibrium solution by “expanding up” along the unstable eigen-direction, hence it is tangent to the unstable eigenspace. In general, unstable manifold expands to the attraction basin of the corresponding limit cycle or attractor. With the introduction of time delay, the two-dimensional unstable manifold of an unstable focus is simulated via expanding solution orbits with restriction condition on the associated foliations. The simulated unstable manifold coincides with the attraction basin of the limit cycle of the delay differential equations.

Topological and geometric hyperbolicity criteria for polynomial automorphisms of

We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$ . Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

Bogdanov–Takens bifurcation of a Holling IV prey–predator model with constant-effort harvesting

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

The Role of Time-Dependent Phase Space Structures in Reaction Dynamics and the No-Recrossing Property of Dividing Surfaces

We analyze benchmark models for reaction dynamics associated with a time-dependent saddle point. Our model allows us to incorporate time dependence of a general form, subject to an exponential growth restriction. Under these conditions, we analytically compute the time-dependent normally hyperbolic invariant manifold; its time-dependent stable and unstable manifolds; and a time-dependent dividing surface that has the no-recrossing property. Consideration of the time dependence of these phase space structures is necessary in order to precisely capture reacting and nonreacting trajectories. Moreover, we show that a time-dependent dividing surface is necessary in order to eliminate recrossing in the time-dependent setting. In other words, if the dividing surface is not time-dependent, recrossing may occur.

Entropies and volume growth of unstable manifolds

Let f be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu $ . We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of $\mu $ in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also discuss extensions to the $C^{1+\alpha },\,\alpha>0$ , case.