The Chaotic Saddle in the Lozi Map, Autonomous and Nonautonomous Versions

In this paper, we prove the existence of a chaotic saddle for a piecewise-linear map of the plane, referred to as the Lozi map. We study the Lozi map in its orientation and area preserving version. First, we consider the autonomous version of the Lozi map to which we apply the Conley–Moser conditions to obtain the proof of a chaotic saddle. Then we generalize the Lozi map on a nonautonomous version and we prove that the first and the third Conley–Moser conditions are satisfied, which imply the existence of a chaotic saddle. Finally, we numerically demonstrate how the structure of this nonautonomous chaotic saddle varies as parameters are varied.

PARTIAL CONTROL OF TRANSIENT CHAOS IN ELECTRONIC CIRCUITS

We present an analog circuit implementation of the novel partial control method, that is able to sustain chaotic transient dynamics. The electronic circuit simulates the dynamics of the one-dimensional slope-three tent map, for which the trajectories diverge to infinity for nearly all the initial conditions after behaving chaotically for a while. This is due to the existence of a nonattractive chaotic set: a chaotic saddle. The partial control allows one to keep the trajectories close to the chaotic saddle, even if the control applied is smaller than the effect of the applied noise, introduced into the system. Furthermore, we also show here that similar results can be implemented on a circuit that simulates a horseshoe-like map, which is a simple extension of the previous one. This encouraging result validates the theory and opens new perspectives for the application of this technique to systems with higher dimensions and continuous time dynamics.

ROLES OF CHAOTIC SADDLE AND BASIN OF ATTRACTION IN BIFURCATION AND CRISIS ANALYSIS

This paper is devoted to the dynamical behavior of a parametrically driven double-well Duffing (PDWD) system. Despite the invariant property of symmetry, this simple model exhibits a large diversity of patterns which can be observed in different situations. The transitions between symmetric forms of system responses often lead to bifurcation or crisis and complicated behaviors, such as the coexistence of different kinds of attractors. The bifurcations and crises are discussed, especially those inside the main periodic window. In particular, the role of chaotic saddles and their intrinsic links with the basin of attraction and transient chaos is studied.

A HYPERCHAOTIC CRISIS

A crisis is investigated in high dimensional chaotic systems by means of generalized cell mapping digraph (GCMD) method. The crisis happens when a hyperchaotic attractor collides with a chaotic saddle in its fractal boundary, and is called a hyperchaotic boundary crisis. In such a case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis, namely, the hyperchaotic attractor is converted into an incremental portion of the hyperchaotic saddle after the collision. This hyperchaotic saddle is an invariant and nonattracting hyperchaotic set. In the hyperchaotic boundary crisis, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary. Two examples of such a hyperchaotic crisis are given in Kawakami map.

ALFVÉN COMPLEXITY

The complex dynamics of Alfvén waves described by the derivative nonlinear Schrödinger equation is investigated. In a region of the parameters space where multistability is observed, this complex system is driven towards an intermittent regime by the addition of noise. The effects of Gaussian and non-Gaussian noise are compared. In the intermittent regime, the Alfvén wave exhibits random qualitative changes in its dynamics as the result of a competition between three attractors and a chaotic saddle embedded in the fractal basin boundary.

Chaos in driven Alfvén systems: unstable periodic orbits and chaotic saddles

The chaotic dynamics of Alfvén waves in space plasmas governed by the derivative nonlinear Schrödinger equation, in the low-dimensional limit described by stationary spatial solutions, is studied. A bifurcation diagram is constructed, by varying the driver amplitude, to identify a number of nonlinear dynamical processes including saddle-node bifurcation, boundary crisis, and interior crisis. The roles played by unstable periodic orbits and chaotic saddles in these transitions are analyzed, and the conversion from a chaotic saddle to a chaotic attractor in these dynamical processes is demonstrated. In particular, the phenomenon of gap-filling in the chaotic transition from weak chaos to strong chaos via an interior crisis is investigated. A coupling unstable periodic orbit created by an explosion, within the gaps of the chaotic saddles embedded in a chaotic attractor following an interior crisis, is found numerically. The gap-filling unstable periodic orbits are responsible for coupling the banded chaotic saddle (BCS) to the surrounding chaotic saddle (SCS), leading to crisis-induced intermittency. The physical relevance of chaos for Alfvén intermittent turbulence observed in the solar wind is discussed.