Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points

Abstract

We show that for periodic non-autonomous discrete dynamical systems, even when a common fixed point for each of the autonomous associated dynamical systems is repeller, this fixed point can became a local attractor for the whole system, giving rise to a Parrondo's dynamic type paradoxPeer ReviewedPostprint (author's final draft

A dynamic Parrondo’s paradox for continuous seasonal systems

“…In fact, ultimately, the proof of Theorem 1 relies on the fact that, near a critical point, the flow of some suitable vector fields are such, up to certain fixed order on the initial conditions, their associated time-1 maps are the ones given in Example 7 of [8]. We recall them in Proposition 11.…”

“…Several dynamical versions of related paradoxes are presented in [4,6,7,8] for discrete non-autonomous dynamical systems. In the first paper the authors combine periodically one-dimensional maps f 1 and f 2 to give rise to chaos or order.…”

“…In this last reference also are shown alternating systems with regular (integrable) dynamics obtained by alternating an integrable map and a numerically chaotic one. In [8] we study a local problem, but in any dimension.…”

“…Theorem 1 is a consequence of the following two results. The first one is proved in [8] but for completeness we include a sketch of its proof. The second one is a consequence of the results in the previous section.…”

“…In [8], both the Birkhoff and the Birkhoff stability constants of F 1 and F 2 are computed obtaining that B 1 (F 1 ) = − 1 2 − 11 2 i and B 1 (F 2 ) = − 1 2 + √ 3 2 i. So V 1 (F j ) = − 1 2 < 0 for j = 1, 2, and the origin is LAS for both maps F 1 and F 2 .…”