Chaotic attractors of a locally conservative hyperbolic map with overlap (original) (raw)
Related papers
Unstable periodic orbits and the dimensions of multifractal chaotic attractors
Physical review. A, 1988
The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily Sne-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and chaotic repeBers are considered.
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000
For a family of logisticlike maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase-space volume occupied by the ensemble W(t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflection of the map near its extremal point. Further, we investigate the temporal evolution of W(t) for the circle map whose critical attractor is dense. In this case, we found W(t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of nonextensive Tsallis entropies.
Phys Rev E, 2000
For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble W(t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of W(t) for the circle map whose critical attractor is dense. In this case, we found W(t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of nonextensive Tsallis entropies.
Multifractality and nonextensivity at the edge of chaos of unimodal maps
Physica A: Statistical Mechanics and its Applications, 2004
We examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity z > 1. First we determine analytically the sensitivity to initial conditions ξ t. Then we consider a renormalization group (RG) operation on the partition function Z of the multifractal attractor that eliminates one half of the multifractal points each time it is applied. Invariance of Z fixes a length-scale transformation factor 2 −η in terms of the generalized dimensions D β. There exists a gap ∆η in the values of η equal to λ q = 1/(1 − q) = D −1 ∞ − D −1 −∞ where λ q is the q-generalized Lyapunov exponent and q is the nonextensive entropic index. We provide an interpretation for this relationship-previously derived by Lyra and Tsallis-between dynamical and geometrical properties.
Dual multifractal structure in hyperchaotic attractors
arXiv: Chaotic Dynamics, 2016
In the context of chaotic dynamical systems with exponential divergence of nearby trajectories in phase space, hyperchaos is defined as a state where there is divergence or stretching in at least two directions during the evolution of the system. Hence the detection and characterization of a hyperchaotic attractor is usually done using the spectrum of Lyapunov Exponents (LEs) that measure this rate of divergence along each direction. Though hyperchaos arise in different dynamical situations and find several practical applications, a proper understanding of the geometric structure of a hyperchaotic attractor still remains an unsolved problem. In this paper, we present strong numerical evidence to suggest that the geometric structure of a hyperchaotic attractor can be characterized using a multifractal spectrum with two superimposed components. In other words, apart from developing an extra positive LE, there is also a structural change as a chaotic attractor makes a transition to the...
Singular-hyperbolic attractors are chaotic
2005
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a uuu-Gibbs state and an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.
Parametric characterisation of a chaotic attractor using the two scale Cantor measure
Physica D-nonlinear Phenomena, 2009
A chaotic attractor is usually characterised by its multifractal spectrum which gives a geometric measure of its complexity. Here we present a characterisation using a minimal set of independent parameters which is uniquely determined by the underlying process that generates the attractor. The method maps the f(α)f(α) spectrum of a chaotic attractor on to that of a general two scale Cantor measure. We show that the mapping can be done in practice with reasonable accuracy for many of the standard chaotic attractors. In order to implement this procedure, we also propose a generalisation of the standard equations for the two scale Cantor set in one dimension to that in higher dimensions. Another interesting result we have obtained both theoretically and numerically is that, the f(α)f(α) characterisation gives information only up to two scales, even when the underlying process generating the multifractal involves more than two scales.
Extreme fractal structures in chaotic mechanical systems: riddled basins of attraction
Journal of Physics: Conference Series, 2010
Chaotic dynamical systems with certain phase space symmetries may exhibit riddled basins of attraction, which can be viewed as extreme fractal structures in the sense that, regardless of how small is the uncertainty in the determination of an initial condition, we cannot decrease the fraction of such points that are certain to converge to a given attractor. We investigate a mechanical system exhibiting riddled basins of attraction: a particle under a two-dimensional potential with friction and time-periodic forcing. The verification of riddling is made by checking its mathematical requirements through computation of finite-time Lyapunov exponents as well as by scaling laws describing the fine structure of basin filaments densely intertwined in phase space.
Composition of Chaotic Maps with an Invariant Measure
We generate new hierarchy of many-parameter family of maps of the interval [0, with an invariant measure, by composition of the chaotic maps of reference . Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent, of these maps analytically, where the results thus obtained have been approved with numerical simulation. In contrary to the usual one-dimensional maps and similar to the maps of reference [1], these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor at certain region of parameters values, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at these values of parameter whose Lyapunov characteristic exponent begins to be positive.