Unimodal Maps Research Articles

Reverse bifurcations in a unimodal queueing model

We present a family of unimodal maps, arising from a simple queueing model, which exhibits reverse bifurcations. We compare and contrast this with bifurcations occurring in the well-known logistic family of unimodal quadratic maps. Throughout this study, graphics generated via numerical simulations provide key insights.

On Hausdorff Dimension of Unimodal Attractors

There exists a universal constant σ<1 such that every attractor of every C 4 unimodal map with a non-degenerate critical point is an analytic manifold or its Hausdorff dimension is equal to or less than σ.

Decay of geometry for unimodal maps: An elementary proof

We prove that a nonrenormalizable smooth unimodal interval map with critical order between 1 and 2 displays decay of geometry, by an elementary and purely real argument. This completes a real approach to Milnor's attractor problem for smooth unimodal maps with critical order not greater than 2.

Symbolic Dynamics of Coupled Map Lattices

We present a method to reduce the [FORMULA: SEE TEXT] dynamics of coupled map lattices (CMLs) of N invertibly coupled unimodal maps to a sequence of N-bit symbols. We claim that the symbolic description is complete and provides for the identification of all fixed points, periodic orbits, and dense orbits as well as an efficient representation for studying pattern formation in CMLs. We give our results for CMLs in terms of symbolic dynamical concepts well known for one-dimensional chaotic maps, including generating partitions, Gray orderings, and kneading sequences. An example utilizing coupled quadratic maps is given.

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.

On unimodal maps with critical order 2+ε

On unimodal maps with critical order 2+ε

On hausdorff dimension of some cantor attractors

We study what happens with the dimension of Feigenbaum-like attractors of smooth unimodal maps as the order of the critical point grows.

Second eigenvalue of transition matrix associated to iterated maps

Abstract We study the properties of the second eigenvalue of the transition matrix in unimodal and bimodal map’s families. We use in particular the relation with the mixing rate and apply this to classify families of isentropic bimodal maps. We present also a study which show the convergence of this quantity in those families.

Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves

We study the structure of the 2D bifurcation diagram for a two-parameter family of piecewise smooth unimodal maps f with one break point. Analysing the parameters of the normal form for the border-collision bifurcation of an attracting n-cycle of the map f, we describe the possible kinds of dynamics associated with such a bifurcation. Emergence and role of border-collision bifurcation curves in the 2D bifurcation plane are studied. Particular attention is paid also to the curves of homoclinic bifurcations giving rise to the band merging of pieces of cyclic chaotic intervals.

Long tailed maps as a representation of mixed mode oscillatory systems

Mixed mode oscillatory (MMO) systems are known to exhibit generic features such as the reversal of period doubling sequences and crossover to period adding sequences as bifurcation parameters are varied. In addition, they exhibit a nearly one dimensional unimodal Poincare map with a long tail. The numerical results of a map with a unique critical point (map-L) show that these dynamical features are reproduced. We show that a few generic conditions extracted from the map-L are adequate to explain the reversal of period doubling sequences and crossover to period adding sequences. We derive scaling relations that determine the parameter widths of the dominant windows of periodic orbits sandwiched between two successive states of RL k sequence and verify the same with the map-L. As the conditions used to derive the scaling relations do not depend on the form of map, we suggest that the analysis is applicable to a family of two parameter one dimensional maps that satisfy these conditons.

Analyticity of the susceptibility function for unimodal Markovian maps of the interval

We study the expression (susceptibility) where f is a unimodal Markovian map of the interval I, ρ = ρf is the corresponding absolutely continuous invariant measure and A is a C1 function defined on I. We show that Ψ(λ) is analytic near λ = 1, where Ψ(1) is formally the derivative of with respect to f in the direction of the vector field X.

Noncommutative topological dynamics

We study noncommutative dynamical systems associated to unimodal and bimodal maps of the interval. To these maps we associate subshifts and the correspondent AF-algebras and Cuntz–Krieger algebras. As an example we consider systems having equal topological entropy log(1 + /), where / is the golden number, but distinct chaotic behavior and we show how a new numerical invariant allows to distinguish that complexity. Finally, we give a statistical interpretation to the topological numerical invariants associated to bimodal maps. � 2005 Elsevier Ltd. All rights reserved.

Statistical Properties of Unimodal Maps

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.

The phase-modulated logistic map

We study the logistic mapping with the nonlinearity parameter varied through a delayed feedback mechanism. This history dependent modulation through a phaselike variable offers an enhanced possibility for stabilization of periodic dynamics. Study of the system as a function of nonlinearity and modulation parameters reveals new phenomena: In addition to period-doubling and tangent bifurcations, there can be bifurcations where the period increases by unity. These are extensions of crises that arise in nonlinear dynamical systems. Periodic orbits in this system can be systematized via the kneading theory, which in the present case extends the analysis of Metropolis, Stein, and Stein for unimodal maps.

On complete chaotic maps with tent-map-like structures

A unimodal map f : [0, 1] → [0, 1] is said to be complete chaotic if it is both ergodic and chaotic in a probabilistic sense so as to preserve an absolutely continuous invariant measure. The sufficient conditions are provided to construct complete chaotic maps with the tent-map-like structures, that is, f(x) = 1 − ∣1 − 2g(x)∣, where g is an one-to-one onto map defined on [0, 1]. The simplicity and analytical characteristics of such chaotic maps simplify the calculations of various statistical properties of chaotic dynamics.

Nonmonotonicity of Kneading Invariants in a Family of Kinked Maps

We study monotonicity properties of the kneading invariant for one-parameter families of piecewise-linear unimodal maps and prove a theorem on the violation of monotonicity of the kneading invariant for maps that are symmetric and convex and consist of four linear pieces. The fact that such maps cannot be approximated by smooth mappings with negative Schwarzian is proved using a dynamics argument.

Dynamics and Universality of Unimodal Mappings with Infinite Criticality

We consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.

The complexity of Fibonacci-like kneading sequences

The Fibonacci(-like) unimodal maps that have been studied in recent years give rise to a zero-entropy minimal subshift on two symbols, generated by the kneading sequence. In this paper we computed the word-complexity of such subshifts exactly.

Reduction of a model of an excitable cell to a one-dimensional map

We use qualitative methods for singularly perturbed systems of differential equations and the principle of averaging to compute the first return map for the dynamics of a slow variable (calcium concentration) in the model of an excitable cell. The bifurcation structure of the system with continuous time endows the map with distinct features: it is a unimodal map with a boundary layer corresponding to the homoclinic bifurcation in the original model. This structure accounts for different periodic and aperiodic regimes and transitions between them. All parameters in the discrete system have biophysical meaning, which allows for precise interpretation of various dynamical patterns. Our results provide analytical explanation for the numerical studies reported previously.

Statistical properties of unimodal maps: the quadratic family

We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step to get the same results for more general families of unimodal maps.