Piecewise Continuous Maps Research Articles

The peak-end rule and its dynamic realization through differential equations with maxima *

In the 1990s, after a series of experiments, the behavioural psychologist and economist Daniel Kahneman and his colleagues formulated the following peak-end evaluation rule: the remembered utility of pleasant or unpleasant episodes is accurately predicted by averaging the peak (most intense value) of instant utility (or disutility) recorded during an episode and the instant utility recorded near the end of the experience (Kahneman et al 1997 Q. J. Econ. 112 375–405). Based on this rule, we propose a mathematical model for the time evolution of the experienced utility function given by the scalar differential equation where f represents exogenous stimuli, h is the maximal duration of the experience, and are some averaging weights. In this work, we study equation and show that, for a range of parameters and a periodic sine-like term f, the dynamics of can be completely described in terms of an associated one-dimensional dynamical system generated by a piece-wise continuous map from a finite interval into itself. We illustrate our approach with two representative examples. In particular, we show that the utility u(t) (e.g. ‘happiness’, interpreted as hedonic utility) can exhibit chaotic behaviour.

Chaos of the 2D linear hyperbolic equation with general van der Pol type boundary condition

This paper is concerned with the complex dynamics of the initial-boundary value problem of a 2D linear hyperbolic partial differential equation (PDE), where the parameter α that appeared in the general van der Pol type boundary condition is given by α∈R. The whole real line is divided into three intervals of the parameter to discuss the dynamics. The existence of chaos is first established in the sense of the exponential growth of total variation when the parameter locates in the central interval, which allows it to be positive, negative, or zero. By analyzing the chaotic dynamics of the piecewise continuous map induced by the hyperbolic PDE, such a PDE is further rigorously proved to be chaos in the interval of the positive parameter that is to the right of the central interval. Finally, the asymptotic behaviors of the hyperbolic PDE are systematically presented in the rest of the whole real line; more precisely, the hyperbolic PDE is either globally stable or unbounded.

A higher-dimensional generalization of the Lozi map: bifurcations and dynamics

We generalize the two-dimensional Lozi map in order to systematically obtain piecewise continuous maps in three and higher dimensions. Similar to higher dimensional generalizations of the related Hénon map, these higher dimensional Lozi maps support hyperchaotic dynamics. We carry out a bifurcation analysis and investigate the dynamics through both numerical and analytical means. The analysis is extended to a sequence of approximations that smooth the discontinuity of the derivatives in the Lozi map.

The hidden unstable orbits of maps with gaps

Piecewise-continuous maps consist of smooth bran- ches separated by jumps, i.e. isolated discontinui- ties. They appear not to be constrained by the same rules that come with being continuous or differentiable, able to exhibit period incrementing and period adding bifurcations in which branches of attractors seem to appear ‘out of nowhere’, and able to break the rule that ‘period three implies chaos’. We will show here that piecewise maps are not actually so free of the rules governing their continuous cousins, once they are recognized as containing numerous unstable orbits that can only be found by explicitly including the ‘gap’ in the map’s definition. The addition of these ‘hidden’ orbits—which possess an iterate that lies on the discontinuity—bring the theory of piecewise-continuous maps closer to continuous maps. They restore the connections between branches of stable periodic orbits that are missing if the gap is not fully accounted for, showing that stability changes must occur in discontinuous maps via stability changes not so different to smooth maps, and bringing piecewise maps back under the powerful umbrella of Sharkovskii’s theorem. Hidden orbits are also vital for understanding what happens if the discontinuity is smoothed out to render the map continuous and/or differentiable.

Invariant measures for interval translations and some other piecewise continuous maps

We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided all discontinuity points are wandering.

Invariant measures for piecewise continuous maps

We say that f:[0,1]→[0,1] is a piecewise continuous interval map if there exists a partition 0=x0<x1<⋯<xd<xd+1=1 of [0,1] such that f|(xi−1,xi) is continuous and the lateral limits w0+=limx→0+⁡f(x), wd+1−=limx→1−⁡f(x), wi−=limx→xi−⁡f(x) and wi+=limx→xi+⁡f(x) exist for each i. We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. We also prove that every injective piecewise continuous interval map with no connections and no periodic orbits is topologically semiconjugate to an interval exchange transformation.

Topological conjugacy of piecewise monotonic functions of nonmonotonicity height ≥1

The conjugacy problem is an important topic in the theory of dynamical systems and functional equations. In this paper, we investigate a class of piecewise monotone and continuous maps with nonmonotonicity height ≥1. We give a sufficient and necessary condition under which any two of these maps are topologically conjugate, and construct a topological conjugacy with an extension method if such a conjugacy exists.

Calculation of homoclinic and heteroclinic orbits in 1D maps

Homoclinic orbits and heteroclinic connections are important in several contexts, in particular for a proof of the existence of chaos and for the description of bifurcations of chaotic attractors. In this work we discuss an algorithm for their numerical detection in smooth or piecewise smooth, continuous or discontinuous maps. The algorithm is based on the convergence of orbits in backward time and is therefore applicable to expanding fixed points and cycles. For simplicity, we present the algorithm using 1D maps.

Embedding the symbolic dynamics of Lorenz maps

AbstractNecessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

Chaotic Behavior in Piecewise Continuous Maps

Chaotic Behavior in Piecewise Continuous Maps

The space of ω-limit sets of piecewise continuous maps of the interval

According to a well-known result, the collection of all ω-limit sets of a continuous map of the interval equipped with the Hausdorff metric is a compact metric space. In this paper, a similar result is proved for piecewise continuous maps with finitely many points of discontinuity, if the points of discontinuity are not periodic for any variant of the map. A variant of f is a map g coinciding with f at any point of continuity and being continuous from one side at any point of discontinuity. It is also shown that ω-limit sets of these maps are locally saturating, another property known for continuous maps. However, contrary to the situation for continuous maps, there are piecewise continuous maps having locally saturating sets which are not ω-limit sets. A condition implying that a locally saturating set is an ω-limit set is presented.

Lifting measures to inducing schemes

AbstractIn this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [Y. Pesin and S. Senti. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J.5(3) (2005), 669–678; Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemes—which hold for many known examples—any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [J. Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math.112 (1999), 357–380], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [H. Bruin. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys.168(3) (1995), 571–580] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multi-modal maps) and for some multi-dimensional maps.

Dynamical Effects of Missed Switching in Current-Mode Controlled dc–dc Converters

Past investigations on nonlinear phenomena in DC-DC converters assume ideal switching and ignore the practical issue of switching ripples, for which the sampled data models yield piecewise smooth but continuous maps. In this brief, we show that the unavoidable nonidealities in the switching result in discontinuity in the map, which drastically changes the bifurcation structures. We demonstrate these effects experimentally, and develop simple one-dimensional models that account for switching delay and transients, and can predict the bifurcations in such systems with reasonable accuracy. Using the recent developments in bifurcation theory, we derive the limiting conditions for a reliable period-1 operation when these effects are considered.

Coding information and problems of storage in dynamical systems

A method of storage of information in one-dimensional piecewise continuous maps as proposed in Rouabhi [1] is considered. In this report questions in examining the feasibility of information coding process with recognition and retrieving are discussed and analyzed, an approach or an extension is proposed.

Every nonsingular C1 flow on a closed manifold of dimension greater than two has a global transverse disk

We prove three results about global cross sections which are disks, henceforth called global transverse disks. First we prove that every nonsingular (fixed point free) C 1 flow on a closed (compact, no boundary) connected manifold of dimension greater than 2 has a global transverse disk. Next we prove that for any such flow, if the directed graph G h has a loop then the flow does not have a closed manifold which is a global cross section. This property of G h is easy to read off from the first return map for the global transverse disk. Lastly, we give criteria for an “M-cellwise continuous” (a special case of piecewise continuous) map h :D 2→D 2 that determines whether h is the first return map for some global transverse disk of some flow ϕ. In such a case, we call ϕ the suspension of h.

Horseshoes in piecewise continuous maps

This paper presents a framework on existence of horseshoes in piecewise continuous maps, which is more applicable to practical problems.

Electrical activity of a neuron modeled by piecewise continuous maps

We suggest to model the electrical activity of an isolated neuron by using piecewise linear stochastic and piecewise continuous dynamic maps based on phenomenological notions about neuron dynamics. It is demonstrated that the proposed models provide for time series qualitatively similar to those observed in experiments with biological neurons. The possibility of using these models for constructing coupled neuron systems is discussed.

Sаmе peculiarities оf synchronization under changing thе connection structure in small model neuron ensembles, described by piecewise continuous maps and piecewise discontinuous maps

The piecewise-linear stochastic map, the piecewise-continuous dynamic map and piecewise-discontinuous dynamic map, for electrical activity modeling of neuron are proposed. These maps are constructed using the phenomenological notions of biological neuron dynamics. Advantages and limitations of the proposed maps are discussed. The behaviour оf small model neuron ensembles оn the parameters plane is investigated. Considerable dependence between the synchronization degree and the topology of neuron ensemble is shown.

A polynomial bound for the lap number

In this note, we show a polynomial bound for the growth of the lap number of a piecewise monotone and piecewise continuous interval map with finitely many periodic points. We use Milnor and Thurston’s kneading theory with the coordinates of Baladi and Ruelle, which are useful for extending the theory to the non continuous case.

Two applications of noninvertible maps in communication and information storage

Two applications of discrete models in the form of noninvertible maps are presented. The first one is related to the now popular problem of communicating via chaos synchronization in the signal processing field. The second topic concerns a method of information storage in periodic solutions (called cycles) of piecewise continuous noninvertible maps. A procedure for extracting the stored information from an incomplete or a noise corrupted one, by using control of chaos from a feedback loop is proposed.