Self-similarity in periodically forced oscillators

Periodic stimulation of limit cycle oscillations leads to one-dimensional maps f: S¹ → S¹ with two parameters which correspond to the frequency and amplitude of the periodic forcing. Bifurcations are described for the situtation in which f is of topological degree zero. Self-similarity is found in the parameter space.     

Hierarchy of Chaotic Maps with an Invariant Measure and their Compositions

Abstract

We give a hierarchy of many-parameter families of maps of the interval [0, 1] with an invariant measure and using the measure, we calculate Kolmogorov–Sinai entropy of these maps analytically. In contrary to the usual one-dimensional maps 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 space, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at certain values of the parameters.     

Dynamics of simple one-dimensional maps under perturbation

It is known that the one-dimensional discrete maps having single-humped nonlinear functions with the same order of maximum belong to a single class that shows the universal behaviour of a cascade of period-doubling bifurcations from stability to chaos with the change of parameters. This paper concerns studies of the dynamics exhibited by some of these simple one-dimensional maps under constant perturbations. We show that the “universality” in their dynamics breaks down under constant perturbations with the logistic map showing different dynamics compared to the other maps. Thus these maps can be classified into two types with respect to their response to constant perturbations. Unidimensional discrete maps are interchangeably used as models for specific processes in many disciplines due to the similarity in their dynamics. These results prove that the differences in their behaviour under perturbations need to be taken into consideration before using them for modelling any real process.     

Local and global statistical properties of deterministic chaos

Summary Locla and global statistical properties of a class of one-dimensional dissipative chaotic maps and a class of 2-dimensional conservative hyperbolic maps are investigated. This is achieved by considering the spectral properties of the Perron-Frobenius operator (the evolution operator for probability densities) acting on two different types of function space. In the first case, the function space is piecewise analytic, and includes functions having support over local regions of phase space. In the second case, the function space essentially consists of functions which are “globally? analytic,i.e. analytic over the given systems entire phase space. Each function space defines a space of measurable functions or observables, whose statistical moments and corresponding characteristic times can be exactly determined. Paper presented at the International Workshop ?Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena?, Elba, 5–10 June 1994.     

Stochastic Stability¶for Contracting Lorenz Maps and Flows

In a previous work [M], we proved the existence of absolutely continuous invariant measures for contracting Lorenz-like maps, and constructed Sinai–Ruelle–Bowen measures f or the flows that generate them. Here, we prove stochastic stability for such one-dimensional maps and use this result to prove that the corresponding flows generating these maps are stochastically stable under small diffusion-type perturbations, even though, as shown by Rovella [Ro], they are persistent only in a measure theoretical sense in a parameter space. For the one-dimensional maps we also prove strong stochastic stability in the sense of Baladi and Viana[BV]. Received: 24 February 1999 / Accepted: 7 January 2000     

Riddled basins of attraction for synchronized type-I intermittency

Chaotic motion restricted to an invariant subspace of total phase space may be associated with basins of attraction that are riddled with holes belonging to the basin of another limiting state. We study the emergence of such basins for a system of two coupled one-dimensional maps, each exhibiting type-I intermittency.     

若干二维映象中V型阵发层流相区的特征

通过几个实例的分析,说明二维映象中V型阵发的层流相所占据的相空间区域常常具有一维特征.这些一维相空间区域对应于已经由边界碰撞分岔消失的周期点的“鬼魂”留下的一段稳定流形.它们构成类似于一维映象中阵发发生后层流相迭代通过的“隧道”.隧道的开口有明确的位置.V型阵发的湍流相所对应相空间区域具有二维特征.解析讨论了这些特征的形成机制. 关键词： V型阵发 层流相 流形     

Some flesh on the skeleton: The bifurcation structure of bimodal maps

One of the main tools in the numerical study of two-parameter families of one-dimensional maps is the drawing of curves in parameter space corresponding to the existence of superstable periodic orbits. We use kneading theory to describe the structure of these sets of curves for the case of maps with at most two turning points. Then we explain how the bifurcation structure hangs on this “skeleton”.     

Electronically-implemented coupled logistic maps

The logistic map is a paradigmatic dynamical system originally conceived to model thediscrete-time demographic growth of a population, which shockingly, shows that discretechaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we designand characterize a simple, low-cost, easy-to-handle, electronic implementation of thelogistic map. In particular, our implementation allows for straightforwardcircuit-modifications to behave as different one-dimensional discrete-time systems. Also,we design a coupling block in order to address the behavior of two coupled maps, although,our design is unrestricted to the discrete-time system implementation and it can begeneralized to handle coupling between many dynamical systems, as in a complex system. Ourfindings show that the isolated and coupled maps’ behavior has a remarkable agreementbetween the experiments and the simulations, even when fine-tuning the parameters with aresolution of ~10^-3. We support these conclusions by comparing the Lyapunovexponents, periodicity of the orbits, and phase portraits of the numerical andexperimental data for a wide range of coupling strengths and map’s parameters.     

Universal Phenomena in Solution Bifurcations of Some Boundary Value Problems

Abstract

We show that for a class of boundary value problems, the space of initial functions can be stratified dependently on the limit behavior (as the time variable tends to infinity) of solutions. Using known results on universal phenomena appearing in bifurcations of one parameter families of one-dimensional maps, we establish that, for certain types of boundary value dependence, a similar quantitative and qualitative universality is also observed in the stratification and bifurcations of solutions.     

Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincare section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking.     

Stabilizing long-period orbits via symbolic dynamics in simple limiter controllers

We present an efficient approach to determine the control parameter of simple limiter controllers by using symbolic dynamics of one-dimensional unimodal maps. By applying addition- and subtraction-symbol rules for generating an admissible periodic sequence, we deal with the smallest base problem of the digital tent map. The proposed solution is useful for minimizing the configuration of digital circuit designs for a given target sequence. With the use of the limiter controller, we show that one-dimensional unimodal maps may be robustly employed to generate the maximum-length shift-register sequences. For an arbitrary long Sarkovskii sequence, the control parameters are analytically given.     

q-Deformed nonlinear maps

Motivated by studies onq-deformed physical systems related to quantum group structures, and by the elements of Tsallis statistical mechanics, the concept ofq-deformed nonlinear maps is introduced. As a specific example, aq-deformation procedure is applied to the logistic map. Compared to the canonical logistic map, the resulting family ofq-logistic maps is shown to have a wider spectrum of interesting behaviours, including the co-existence of attractors — a phenomenon rare in one-dimensional maps.     

Coupled trivial maps

The first nontrivial example of coupled map lattices that admits a rigorous analysis in the whole range of the strength of space interactions is considered. This class is generated by one-dimensional maps with a globally attracting superstable periodic trajectory that are coupled by a diffusive nearest-neighbor interaction.     

Self-similarities of periodic structures for a discrete model of a two-gene system

We report self-similar properties of periodic structures remarkably organized in the two-parameter space for a two-gene system, described by two-dimensional symmetric map. The map consists of difference equations derived from the chemical reactions for gene expression and regulation. We characterize the system by using Lyapunov exponents and isoperiodic diagrams identifying periodic windows, denominated Arnold tongues and shrimp-shaped structures. Period-adding sequences are observed for both periodic windows. We also identify Fibonacci-type series and Golden ratio for Arnold tongues, and period multiple-of-three windows for shrimps.     

On the control of complex dynamic systems

A method is described for the limited control of the dynamics of systems which generally have several dynamic attractors. associated either with maps or first order ordinary differential equations (ODE) in ⁿ. The control is based on the existence of ‘convergent’ regions, C_C(k = 1,2,…), in the phase space of such systems, where there is ‘local convergence’ of all nearby orbits. The character of the control involves the ‘entrainment’ and subsequent possible ‘migration’ of the experimental system from one attractor to another. Entrainment means that lim_{t > → ∞} |x(t) − g(t)| = 0, where is the system's controlled dynamics, and the goal-dynamics, g(t) ε G_k, has any topological form but is limited dynamically and to regions of phase space, G_k, contained in some C_k, G_k C_k. The control process is initiated only when the system enters a ‘basin of entrainment’, BE_k G_k, associated with the goal-region G_k. Aside from this ‘macroscopic’ initial-state information about the system, no further feedback of dynamic information concerning the response of the system is required. The experimental reliability of the control requires that the regions, BE_k, be convex regions in the phase space, which can apparently be assured if G_k C_k. Simple illustrations of these concepts are given, using a general linear and a piecewise-linear ODE in . In addition to these entrainment-goals, ‘migration-goal’ dynamics is introduced, which intersects two convergent regions G ∩ C_j ≠ , G ∩ C_j ≠ (i ≠ j), and permits transferring the dynamics of a system from one attractor to another, or from one convergent region to another. In the present study these concepts are illustrated with various one-dimensional maps involving one or more attractors and convergent regions. Several theorems concerning entrainment are derived for very general, continuous one-dimensional maps. Sufficient conditions are also established which ensure ‘near-entrainment’ for a system, when the dynamic model of the system is not exactly known. The applications of these concepts to higher dimensional maps and flows will be presented in subsequent studies.     

Entropic Bounds on Semiclassical Measures for Quantized One-Dimensional Maps

Quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic Hamiltonian systems are equidistributed in phase space. This, however, does not prohibit existence of exceptional sequences which might converge to different (non-Liouville) classical invariant measures. It has been recently shown by N. Anantharaman and S. Nonnenmacher in [20,21] (with H. Koch) that for Anosov geodesic flows the metric entropy of any semiclassical measure μ must satisfy a certain bound. This remarkable result seems to be optimal for manifolds of constant negative curvature, but not in the general case, where it might become even trivial if the (negative) curvature of the Riemannian manifold varies a lot. It has been conjectured by the same authors, that in fact, a stronger bound (valid in the general case) should hold. In the present work we consider such entropic bounds using the model of quantized piecewise linear one-dimensional maps. For a certain class of maps with non-uniform expansion rates we prove the Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps we are able to construct some explicit sequences of eigenstates which saturate the bound. This demonstrates that the conjectured bound is actually optimal in that case.     

Classification of one-dimensional chaotic models

Classifying chaotic maps using the relation between a map and its conjugate basic map with uniform invariant distribution is suggested. It is shown that every symmetric one-dimensional chaotic map with two monotonic branches is topologically equivalent to a tent map or a Bernoulli shift. An algorithm for finding a function conjugating two maps is formulated.     

Fermionic quasiparticle representation of Tomonaga-Luttinger Hamiltonian

We find a unitary operator which asymptotically diagonalizes the Tomonaga-Luttinger Hamiltonian of one-dimensional spinless electrons. The operator performs a Bogoliubov rotation in the space of electron-hole pairs. If bare interaction of the physical electrons is sufficiently small this transformation maps the original Tomonaga-Luttinger system on a system of free fermionic quasiparticles. Our representation is useful when the electron dispersion deviates from linear form. For such situation we obtain non-perturbative results for the electron gas free energy and the density-density propagator.     

Geometry of financial markets—Towards information theory model of markets

Most parameters used to describe states and dynamics of financial market depend on proportions of the appropriate variables rather than on their actual values. Therefore, projective geometry seems to be the correct language to describe the theater of financial activities. We suppose that the objects of interest of agents, called here baskets, form a vector space over the reals. A portfolio is defined as an equivalence class of baskets containing assets in the same proportions. Therefore portfolios form a projective space. Cross ratios, being invariants of projective maps, form key structures in the proposed model. Quotation with respect to an asset Ξ (i.e. in units of Ξ) is given by linear maps. Among various types of metrics that have financial interpretation, the min-max metric on the space of quotations can be introduced. This metric has an interesting interpretation in terms of rates of return. It can be generalized so that to incorporate a new numerical parameter (called temperature) that describes agent's lack of knowledge about the state of the market. In a dual way, a metric on the space of market quotation is defined. In addition, one can define an interesting metric structure on the space of portfolios/quotation that is invariant with respect to hyperbolic (Lorentz) symmetries of the space of portfolios. The introduced formalism opens new interesting and possibly fruitful fields of research.