Dynamic Analysis of a Lü Model in Six Dimensions and Its Projections

In this article, extended complex Lü models (ECLMs) are proposed. They are obtained by substituting the real variables of the classical Lü model by complex variables. These projections, spanning from five dimensions (5D) and six dimensions (6D), are studied in their dynamics, which include phase spaces, calculations of eigenvalues and Lyapunov’s exponents, Poincaré maps, bifurcation diagrams, and related analyses. It is shown that in the case of a 5D extension, we have obtained chaotic trajectories; meanwhile the 6D extension shows quasiperiodic and hyperchaotic behaviors and it exhibits strange nonchaotic attractor (SNA) features.

NOTE ON STRANGE NONFRACTAL ATTRACTORS

We present a case of a strange nonchaotic attractor (SNA) which is also nonfractal and continuous. This provides a counterexample to the widely extended assumption about the intrinsic fractal nature of any SNA. We also show that the most useful techniques to characterize the SNA fail for this case.

RENORMALIZATION GROUP APPROACH TO THE EFFECT OF NOISE AT THE BIRTH OF A STRANGE NONCHAOTIC ATTRACTOR

Scaling regularities associated with additive noise are examined in a model of the pitch-fork bifurcation map with multiplicative quasiperiodic driving (Grebogi et al., Physica D13, 261) with the golden-mean frequency ratio at the birth of a strange nonchaotic attractor (SNA). This case of the onset of SNA termed as the blowout bifurcation route was discussed in the context of realistic systems governed by non-autonomous differential equations (Yalçynkaya and Lai, Phys. Rev. Lett., 77, 5039). Our method taking into the account of noise is based on renormalization group (RG) analysis of the birth of SNA (Kuznetsov et al., Phys. Rev. E51, 1629) with application of an appropriate generalization of the approach of Crutchfield et al. (Phys. Rev. Lett., 46, 933) and Shraiman et al. (Phys. Rev. Lett., 46, 935) originally developed for the period doubling transition to chaos. A constant γ=7.4246 is evaluated that determines the scaling law regarding the intensity of noise: A decrease of the noise amplitude by this factor allows resolving one more level of the fractal-like structure associated with the characteristic time scale which is increased by a factor of [Formula: see text]. Numeric results demonstrating evidence of the expected regularities are presented, e.g. portraits of the noisy attractors in different scales.