We call a chaotic dynamical system pseudo-deterministic when it does not produce numerical, or pseudo-trajectories that stay close, or shadow chaotic true trajectories, even though the model equations are strictly deterministic. In this case, single chaotic trajectories may not be meaningful, and only statistical predictions, at best, could be drawn on the model, like in a stochastic system. The dynamical reason for this behavior is nonhyperbolicity characterized either by tangencies of stable and unstable manifolds or by the presence of periodic orbits embedded in a chaotic invariant set with a different number of unstable directions. We emphasize herewith the latter by studying a low-dimensional discrete-time model in which the phenomenon appears due to a saddle-repeller bifurcation. We also investigate the behavior of the finite-time Lyapunov exponents for the system, which quantifies this type of nonhyperbolicity as a system parameter evolves past a critical value. We argue that the effect of unstable dimension variability is more intense when the invariant chaotic set of the system loses transversal stability through a blowout bifurcation.

Computer simulations of partial differential equations of mathematical physics typically lead to some kind of high-dimensional dynamical system. When there is chaotic behavior we are faced with fundamental dynamical difficulties. We choose as a paradigm of such high-dimensional system a kicked double rotor. This system is investigated for parameter values at which it is strongly non-hyperbolic through a mechanism called unstable dimension variability, through which there are periodic orbits embedded in a chaotic attractor with different numbers of unstable directions. Our numerical investigation is primarily based on the analysis of the finite-time Lyapunov exponents, which gives us useful hints about the onset and evolution of unstable dimension variability for the double rotor map, as a system parameter (the forcing amplitude) is varied.

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ABSTRACT. In [4], S. Elaydi obtained a characterization of the stability of the null solution of the Volterra difference equationby localizing the roots of its characteristic equationThe assumption that (a n ) ∈ 1 was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if the ratio R of convergence of the power series of the previous equation equals one. In fact, when R = 1, this characterization conflicts with a result obtained by Erdös et al. in [8]. Here, we analyze the R = 1 case and show that some parts of that characterization still hold. Furthermore, studies on stability for the R < 1 case are presented. Finally, we study some results related to stability via finite approximation.

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