Improving the Complexity of the Lorenz Dynamics

A new four-dimensional, hyperchaotic dynamic system, based on Lorenz dynamics, is presented. Besides, the most representative dynamics which may be found in this new system are located in the phase space and are analyzed here. The new system is especially designed to improve the complexity of Lorenz dynamics, which, despite being a paradigm to understand the chaotic dissipative flows, is a very simple example and shows great vulnerability when used in secure communications. Here, we demonstrate the vulnerability of the Lorenz system in a general way. The proposed 4D system increases the complexity of the Lorenz dynamics. The trajectories of the novel system include structures going from chaos to hyperchaos and chaotic-transient solutions. The symmetry and the stability of the proposed system are also studied. First return maps, Poincaré sections, and bifurcation diagrams allow characterizing the global system behavior and locating some coexisting structures. Numerical results about the first return maps, Poincaré cross sections, Lyapunov spectrum, and Kaplan-Yorke dimension demonstrate the complexity of the proposed equations.

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.

ON THE ROLE OF CHAOTIC SADDLES IN GENERATING CHAOTIC DYNAMICS IN NONLINEAR DRIVEN OSCILLATORS

In the paper, the most important common dynamical element underlying the build-up of chaotic responses in nonlinear vibrating systems, i.e. the formation and expansion of invariant non-attracting chaotic sets, so-called chaotic saddles, as a result of transverse intersections of stable and unstable invariant manifolds of particular unstable orbits, is highlighted. Characteristic examples of the resulting multiple aspects of chaotic system behaviors, such as chaotic transient motions, fractal basin boundaries and unpredictability of the final state, are shown and discussed with the use of geometrical interpretation of the results completed by color computer graphics. Numerical study is carried out for two low-dimensional but representative models of nonlinear, strictly dissipative oscillators driven externally by periodic force, i.e. the twin-well Duffing oscillator and the plane pendulum. In particular, it is demonstrated that the formation of chaotic saddles (equivalent to the creation of horseshoes in the system dynamics) is the primary mechanism triggering chaotic transient motions independently of either single or multiple attractors exist. The aspect of formation of chaotic saddles as a result of a sequence of global (homoclinic and heteroclinic) bifurcations, which is useful in establishing criteria for the occurrence of chaotic system behaviors as the control parameter changes, is presented.

SUPER PERSISTENT CHAOTIC TRANSIENTS IN PHYSICAL SYSTEMS: EFFECT OF NOISE ON PHASE SYNCHRONIZATION OF COUPLED CHAOTIC OSCILLATORS

A super persistent chaotic transient is typically induced by an unstable–unstable pair bifurcation in which two unstable periodic orbits of the same period coalesce and disappear as a system parameter is changed through a critical value. So far examples illustrating this type of transient chaos utilize discrete-time maps. We present a class of continuous-time dynamical systems that exhibit super persistent chaotic transients in parameter regimes of positive measure. In particular, we examine the effect of noise on phase synchronization of coupled chaotic oscillators. It is found that additive white noise can induce phase slips in integer multiples of 2π's in parameter regimes where phase synchronization is expected in the absence of noise. The average time durations of the temporal phase synchronization are in fact characteristic of those of super persistent chaotic transients. We provide heuristic arguments for the scaling law of the average transient lifetime and verify it using numerical examples from both the system of coupled Chua's circuits and that of coupled Rössler oscillators. Our work suggests a way to observe super persistent chaotic transients in physically realizable systems.

RELATION BETWEEN UNCERTAINTY EXPONENT AND MEAN LIFETIME OF CHAOTIC TRANSIENT FOR MAP ON ANNULUS

For a map on the annulus, it is tested numerically that, within errors of the calculation of 0.4%, the inverse of the mean lifetime of chaotic transient is equal to the product of the uncertainty exponent and the Lyapunov exponent. The second-order term in the Taylor series expansion for inverse lifetime has no effect within the precision of the calculation.