Distribution Of Finite-time Lyapunov Exponents Research Articles

Intermittency transitions to strange nonchaotic attractors in a quasiperiodically driven duffing oscillator

Different mechanisms for the creation of strange nonchaotic attractors (SNAs) are studied in a two-frequency parametrically driven Duffing oscillator. We focus on intermittency transitions in particular, and show that SNAs in this system are created through quasiperiodic saddle-node bifurcations (type-I intermittency) as well as through a quasiperiodic subharmonic bifurcation (type-III intermittency). The intermittent attractors are characterized via a number of Lyapunov measures including the behavior of the largest nontrivial Lyapunov exponent and its variance, as well as through distributions of finite-time Lyapunov exponents. These attractors are ubiquitous in quasiperiodically driven systems; the regions of occurrence of various SNAs are identified in a phase diagram of the Duffing system.

Characteristic distributions of finite-time Lyapunov exponents.

We study the probability densities of finite-time or local Lyapunov exponents in low-dimensional chaotic systems. While the multifractal formalism describes how these densities behave in the asymptotic or long-time limit, there are significant finite-size corrections, which are coordinate dependent. Depending on the nature of the dynamical state, the distribution of local Lyapunov exponents has a characteristic shape. For intermittent dynamics, and at crises, dynamical correlations lead to distributions with stretched exponential tails, while for fully developed chaos the probability density has a cusp. Exact results are presented for the logistic map, x-->4x(1-x). At intermittency the density is markedly asymmetric, while for "typical" chaos, it is known that the central limit theorem obtains and a Gaussian density results. Local analysis provides information on the variation of predictability on dynamical attractors. These densities, which are used to characterize the nonuniform spatial organization on chaotic attractors, are robust to noise and can, therefore, be measured from experimental data.

Strange nonchaotic attractors in the quasiperiodically forced logistic map

Different mechanisms for the creation of strange nonchaotic dynamics in the quasiperiodically forced logistic map are studied. These routes to strange nonchaos are characterized through the behavior of the largest nontrivial Lyapunov exponent, as well as through the characteristic distributions of finite-time Lyapunov exponents. Strange nonchaotic attractors can be created at a saddle-node bifurcation when the dynamics shows type-I intermittency; this intermittent transition, which is studied in detail, is characterized through scaling exponents. Band-merging crises through which dynamics remains nonchaotic are also studied, and correspondence is made with analogous behavior in the unforced logistic map. Robustness of these phenomena with respect to additive noise is investigated.