Non-hyperbolic Dynamical Systems Research Articles

Anomalous dynamics in symmetric triangular irrational billiards

We identify a symmetry induced mechanism which dominates the long time behaviour in symmetric triangular billiards. We rigorously prove the existence of invariant sets in symmetric irrational billiards on which the dynamics is governed by an interval exchange transformation. Counterintuitively, this property of symmetric irrational billiards is analogous to the case of general rational billiards, and it highlights the non-trivial impact of symmetries in non-hyperbolic dynamical systems. Our findings provide an explanation for the logarithmic subdiffusive relaxation processes observed in certain triangular billiards. In addition we are able to settle a long standing conjecture about the existence of non-periodic and not everywhere dense trajectories in triangular billiards.

Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems

We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We also pose interesting questions for future research.

Orbit-counting in non-hyperbolic dynamical systems

There are well-known analogues of the prime number theorem and Mertens' Theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections. The proof of the dynamical analogue of Mertens' Theorem uses transcendence theory and Dirichlet characters.

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameter-dependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green–Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.

Fractal structures of normal and anomalous diffusion in nonlinear nonhyperbolic dynamical systems.

A paradigmatic nonhyperbolic dynamical system exhibiting deterministic diffusion is the smooth nonlinear climbing sine map. We find that this map generates fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. The measure of these self-similar sets is positive, parameter dependent, and in case of normal diffusion it shows a fractal diffusion coefficient. By using a Green-Kubo formula we link these fractal structures to the nonlinear microscopic dynamics in terms of fractal Takagi-like functions.

Theory of stability regions for a class of nonhyperbolic dynamical systems and its application to constraint satisfaction problems

The concept of a stability region (region of attraction) of nonlinear dynamical systems is widely applied in many fields such as engineering and the sciences. In this paper, we develop a comprehensive theory of the stability region and its boundary for a class of nonhyperbolic dynamical systems. We then apply the theory to develop an effective method for solving constraint satisfaction problems. Several necessary and sufficient conditions for an equilibrium manifold (the generalized concept of an equilibrium point) to lie on the stability boundary of the class of nonhyperbolic dynamical systems are derived. The stability boundary of such nonhyperbolic dynamical systems is completely characterized and shown to consist of the union of the stable manifolds of the equilibrium manifolds on the stability boundary. An effective scheme to estimate the stability region of such a dynamical system by using an energy function is developed. These analytical results are then applied to the development of computational methods to systematically find feasible regions of constraint satisfaction problems. Several numerical examples are given to illustrate the effectiveness of the computational method.

What symbolic dynamics do we get with a misplaced partition?: On the validity of threshold crossings analysis of chaotic time-series

An increasingly popular method of encoding chaotic time-series from physical experiments is the so-called threshold crossings technique, where one simply replaces the real valued data with symbolic data of relative positions to an arbitrary partition at discrete times. The implication has been that this symbolic encoding describes the original dynamical system. On the other hand, the literature on generating partitions of non-hyperbolic dynamical systems has shown that a good partition is non-trivial to find. It is believed that the generating partition of non-uniformly hyperbolic dynamical system connects “primary tangencies”, which are generally not simple lines as used by a threshold crossings. Therefore, we investigate consequences of using itineraries generated by a non-generating partition. We do most of our rigorous analysis using the tent map as a benchmark example, but show numerically that our results likely generalize. In summary, we find the misrepresentation of the dynamical system by “sample-path” symbolic dynamics of an arbitrary partition can be severe, including (sometimes extremely) diminished topological entropy, and a high degree of non-uniqueness. Interestingly, we find topological entropy as a function of misplacement to be devil’s staircase-like, but surprisingly non-monotone.

Universality and scaling in chaotic attractor to chaotic attractor transitions in an optical ring cavity resonator

Abstract A new class of second order phase transitions is identified and characterized for a plane wave intracavity field in an optical ring resonator. In this paper we discuss chaotic attractor to chaotic attractor transitions in a ring cavity laser as an external parameter is varied. We show that the transition is sharply defined and may be classed as a second order phase transition. We obtain scaling laws about the critical point ηc, for the average positive Lyapunov exponent, λ+ |η ηc|−β and the average crisis induced mean lifetime, τ ∼ |η ηc|−γ, where η is the parameter that is varied. Here average means averaged over many initial conditions. Futhermore we find that there is an algebraic relationship between the critical exponents and the correlation dimension D c at the critical point ηc, namely, β + γ + D c = constant. We postulate that this is a universal relationship for second order phase transitions in two-dimensional multiparameter non-hyperbolic dynamical systems and we propose that the Ikeda ring cavity resonator be used to test the relationship experimentally.

Universality and scaling in chaotic attractor to chaotic attractor transitions in an optical ring cavity resonator

A new class of second order phase transitions is identified and characterized for a plane wave intracavity field in an optical ring resonator. In this paper we discuss chaotic attractor to chaotic attractor transitions in a ring cavity laser as an external parameter is varied. We show that the transition is sharply defined and may be classed as a second order phase transition. We obtain scaling laws about the critical point ηc, for the average positive Lyapunov exponent, λ+ |η ηc|−β and the average crisis induced mean lifetime, τ ∼ |η ηc|−γ, where η is the parameter that is varied. Here average means averaged over many initial conditions. Futhermore we find that there is an algebraic relationship between the critical exponents and the correlation dimension D c at the critical point ηc, namely, β + γ + D c = constant. We postulate that this is a universal relationship for second order phase transitions in two-dimensional multiparameter non-hyperbolic dynamical systems and we propose that the Ikeda ring cavity resonator be used to test the relationship experimentally.

Comments on the non-stationary chaos

Non-stationary chaos is a universal phenomenon in non-hyperbolic dynamical systems. Basic problems regarding the non-stationarity are discussed from ergodic-theoretical viewpoints. By use of a simple system, it is shown that “the law of large number” as well as “the law of small number” break down in the non-stationary regime. The non-stationarity in dynamical systems proposes a crucial problem underlying in the transitional region between chance and necessity, where non-observable processes behind reality interplay with observable ones. The incompleteness of statistical ensembles is discussed from the Karamata's theory. Finally, the significance of the stationary/non-stationary interface is emphasized in relation to the universality of 1/ f fluctuations.

Comparison of several generating partitions of the Hénon map

The existence and uniqueness of generating partitions in non-hyperbolic dynamical systems is usually studied in a simple, exemplary system, namely the Henon map at standard parameter values a = 1.4 and b = 0.3. We compare its standard partition with three other binary partitions, which are quite different from the standard partition but also appear to be generating. One of these partitions passes twice through the same orbit of homoclinic tangencies, providing a counterexample to a recent conjecture by Jaeger and Kantz ( J. Phys. A: Math. Gen. 30 L567). Introducing some simple rules to manipulate symbolic sequences, we show how to translate symbolic sequences produced by one partition into sequences produced by the other partitions. This proves that all these partitions are as good approximations to generating partitions as the standard partition. We also construct an infinite number of binary partitions, which are all quite similar to the standard partition, derive their translation rules, and prove the same equivalence. It is not known for sure whether any of these partitions is indeed generating. But if one of them is generating, then they all are.

Dependence of phase transitions on small changes.

In this contribution, the generalized thermodynamic formalism is applied to a nonhyperbolic dynamical system in two comparable situations. The change from one situation to the other is small in the sense that the grammar and the singularities of the system are preserved. For the discussion of the effects generated by this change, the generalized entropy functions are calculated and the sets of the specific scaling functions which reflect the phase transition of the system are investigated. It is found that even under mild variations, this set is not invariant.

A possible scheme for classifying phase transitions in the generalized entropy spectrum of a nonhyperbolic system

In this contribution, the generalized thermodynamic formalism is applied to a nonhyperbolic dynamical system. The system is used to discuss the occurence of combination of phase transitions among different entropy-like scaling functions. For the discussion, the associated generalized entropy function is calculated from an approximation of finite level and for the asymptotic case. From this unifying description of the system, the information contained in the particular spectra as well as the relation among the different spectra can be derived and made visible in an easy way. It is shown how the phase-transition-like behavior leads to the manifestation of a critical line in the surface described by the generalized entropy function which, in turn, determines the occurence of phase transitions for the particular spectra. Furthermore, it is indicated how apparent phase transitions are encountered in approximations of a finite level which do not persist in the asymptotic limit.

Phase transitions in the approximated and asymptotic generalized entropy spectrum of a nonhyperbolic system.

In this contribution, the generalized thermodynamic formalism is applied to a nonhyperbolic dynamical system. The system is used to discuss the occurrence of combinations of phase transitions among different entropylike scaling functions. For the discussion, the associated generalized entropy function is calculated from an approximation of finite level and for the asymptotic case. From this unified description of the system, the information contained in the particular spectra as well as the relation among the different spectra can be elucidated in an elegant way. The phase-transition-like behavior leads to the manifestation of a critical line in the surface described by the generalized entropy function which, in turn, determines the occurrence of phase transitions for the particular spectra. Furthermore, apparent phase transitions are encountered in approximations of a finite level. They do not persist in the asymptotic limit.

Phase Transitions in the Generalized Entropy Spectrum of Nonhyperbolic Dynamical Systems

Abstract In this contribution, the Generalized Thermodynamic Formalism is applied to a nonhyperbolic dynamical system. The associated generalized entropy function is discussed. The model considered is equipped with a generating partition of two symbols; in contrast to analogous hyperbolic models, the support of the generalized entropy function is two-dimensional. Furthermore, the phase-transition- like behavior caused by the nonhyperbolicity of the measure is shown to lead to the manifestation of a critical line in the surface described by the generalized entropy function. It is discussed how the different combinations of phase-transitions in the more specific entropy spectra fx), SG (ε) and ϕ(λ) are determined by the form of the generalized entropy function