Piecewise Smooth Maps Research Articles

Statistical Properties and Decay of Correlations for Interval Maps with Critical Points and Singularities

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the central limit theorem and a vector-valued almost sure invariance principle. We also obtain results on decay of correlations and large deviations.

High-Feedback Operation of Power Electronic Converters

The purpose of this review is to provide a survey of some of the most important bifurcation phenomena that one can observe in pulse-modulated converter systems when operating with high corrector gain factors. Like other systems with switching control, electronic converter systems belong to the class of piecewise-smooth dynamical systems. A characteristic feature of such systems is that the trajectory is “sewed” together from subsequent discrete parts. Moreover, the transitions between different modes of operation in response to a parameter variation are often qualitatively different from the bifurcations we know for smooth systems. The review starts with an introduction to the concept of border-collision bifurcations and also demonstrates the approach by which the full dynamics of the piecewise-linear, time-continuous system can be reduced to the dynamics of a piecewise-smooth map. We describe the main bifurcation structures that one observes in three different types of converter systems: (1) a DC/DC converter; (2) a multi-level DC/DC converter; and (3) a DC/AC converter. Our focus will be on the bifurcations by which the regular switching dynamics becomes unstable and is replaced by ergodic or resonant periodic dynamics on the surface of a two-dimensional torus. This transition occurs when the feedback gain is increased beyond a certain threshold, for instance in Electronics 2013, 2 114 order to improve the speed and accuracy of the output voltage regulation. For each of the three converter types, we discuss a number of additional bifurcation phenomena, including the formation and reconstruction of multi-layered tori and the appearance of phase-synchronized quasiperiodicity. Our numerical simulations are compared with experimentally observed waveforms.

An alternative approach to generalised BV and the application to expanding interval maps

We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where the weighting used in the transfer operator is not better than piecewise Hölder continuous and the partition on which the map is continuous may possess a countable number of elements.For such weighted transfer operators we give upper bounds for both the spectral radius and for the essential spectral radius.

Expectations and industry location: a discrete time dynamical analysis

The new economic geography (NEG) aims to explain long-term patterns in the spatial allocation of industrial activities. It stresses that endogenous economic processes may enlarge small historic differences leading to quite different regional patterns—history matters for the long-term geographical distribution of economic activities. A pivotal element is that productive factors move to another region whenever the anticipated remuneration is higher in that region. Given the long-term nature of NEG analyses and the crucial role of expectations, it is astonishing that most of the existing models assume only naive or myopic expectations. However, a recent stream of the literature in behavioral and experimental economics shows that agents often use expectational heuristics, such as trend extrapolating and trend reverting rules. We introduce such expectations formation hypotheses into a NEG model formulated in discrete time. This modification leads to a system of two nonlinear difference equations (corresponding, in the language of dynamical systems theory, to a 2-dimensional piecewise smooth map) and thus enriches the possible dynamic patterns: with trend extrapolating (reverting) the symmetric equilibrium is less (more) stable; and it may lose stability only via a flip bifurcation (or also via a Neimark–Sacker bifurcation) giving rise to a period-doubling cascade (or also to quasi-periodic orbits). In both cases, complex behavior is possible; multistability, that is, the coexistence of locally stable equilibria, is pervasive; and border-collision bifurcations are also allowed. In this sense, our analysis corroborates some of the basic insights of the NEG.

Visualization of high-dimensional data using two-dimensional self-organizing piecewise-smooth Kohonen maps

To make visualization of high-dimensional data more accurate, we offer a method of approximating two-dimensional Kohonen maps lying in a multiple-dimensional space. Cubic parametric spline-based least-defect surfaces can be used as an approximation function to minimize approximation errors.

Structurally unstable regular dynamics in 1D piecewise smooth maps, and circle maps

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X′=aXif ∣X∣<z, X′=bX if ∣X∣>z, where a and b can take any real value,and may have several applications. We show that its dynamic behaviors are those of a linear rotation:either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X′=F(X) if ∣X∣<z, X′=G(X) if ∣X∣>z is also given, proving the conditions leading to a homeomorphism of the circle.

Analysis of stable periodic orbits in the one dimensional linear piecewise-smooth discontinuous map

In this paper, we consider one dimensional linear piecewise-smooth discontinuous maps. It is well known that stable periodic orbits exist for such maps, in some parameter region. It is also known that the corresponding bifurcation phenomena (termed as period adding bifurcation) exhibit a special structure. In the last couple of years, several authors have analyzed this structure using border collision bifurcation curves and given the characterization for various parameter regions. In this paper, we have analyzed a specific parameter range employing a different approach. We show that this approach enables one to pose some interesting questions like: what is the number of distinct periodic orbits of any given cardinality? We prove that there are precisely ϕ(n) distinct orbits of period n, where ϕ is the Euler’s totient function. We propose an algorithm which calculates the location of fixed points of all these ϕ(n) distinct orbits and gives the precise range of existence of these orbits with respect to the parameters. Further, we show how the amount of computations required to find these ranges of existence can be optimized.

Bifurcation analysis of an inductorless chaos generator using 1D piecewise smooth map

In this work we investigate the dynamics of a one-dimensional piecewise smooth map, which represents the model of a chaos generator circuit. In a particular (symmetric) case analytic results can be given showing that the chaotic region is wide and robust. In the general model only the border collision bifurcation can be analytically determined. However, the dynamics behave in a similar way, leading effectively to robust chaos.

Behaviour of the escape rate function in hyperbolic dynamical systems

For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Hölder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory.

Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map

In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the discontinuous flat top tent map every unstable periodic orbit may become a Milnor attractor. Moreover, it turns out that there exists a strong connection between stable and unstable orbits and that the appearance of a single unstable orbit may cause an infinite number of stable orbits to appear. Based on this connection we provide a more precise explanation of the recently discovered self-similar bifurcation scenario occurring in the discontinuous flat top tent map denoted as the nested period incrementing scenario.

Statistical properties of nonuniformly expanding 1D maps with logarithmic singularities

We study the statistical properties of piecewise smooth maps on a circle, with a finite number of critical and singular points with an unbounded derivative, such that the derivative goes like the inverse of the distance to the singular points. We write down a simple set of conditions, and show that when these conditions are met, there exist an absolutely continuous invariant probability measure with exponential decay of correlations. We also rule out the existence of nontrivial coboundary, and obtain a positive variance in the central limit theorem for any nonconstant Hölder continuous observable. Our results apply to a positive measure set of nonuniformly expanding maps on the circle considered by Takahasi and Wang (2012 Nonlinearity 25 533).

CHARACTERIZATION OF CHAOS SCENARIOS WITH PERIODIC INCLUSIONS FOR ONE CLASS OF PIECEWISE-SMOOTH DYNAMICAL MAPS

In this paper, we study bifurcation scenarios characterized by period-adding cascades with alternating chaos in one class of piecewise-smooth maps (PWS). In this class, the state space is separated in three smooth zones defined by a saturation function. Some power converters controlled by Digital Pulse-Width Modulation (PWM) are physical applications of this class of PWS systems denoted by PWS3. Chaos has virtually been detected and studied in all disciplines, however the characterization problem of chaos scenarios has many open problems, mainly in nonsmooth dynamical systems. Novel bifurcation scenarios have recently been reported such as bandcount adding and bandcount increment scenarios based on the numerical detection of bands (where bands are considered as strongly connected components). However, this approach known as Bandcounter cannot be applied to detect bifurcations in chaos scenarios without crisis bifurcations or to identify topological changes inside of one-band chaos. We have proposed a novel framework named Dynamic Linkcounter approach to characterize chaos and torus breakdown scenarios in PWS systems. In this paper, we report overlapping period-adding cascades interspersed with a dynamic linkcount adding cascade. Each complex dynamic link (CDL) structure is a fingered strange attractor increasing in an arithmetic progression the number of CDL or fingers when a bifurcation parameter is varied. Alternative point of view based on tent-map-like structures is given to illustrate the formation of fingered strange attractors.

Bifurcation and Border Collision Analysis of Voltage-Mode-Controlled Flyback Converter Based on Total Ampere-Turns

Total ampere-turns of the primary and secondary windings of a transformer are taken as one of variables to analyze the bifurcation behaviors and the border collision phenomenon on voltage-mode-controlled flyback converter. Six operation modes which are separated by the two borderlines of capacitor voltage and the three borderlines of total ampere-turns are obtained. A class of explicit piecewise smooth discrete-time maps with six operation modes is derived to describe the dynamics of the system. The numerical calculations are carried out, the bifurcation diagrams and the maximal Lyapunov exponent spectrums with the control period as a parameter are obtained. Then, the paper particularly analyzes the stability and existence conditions of main periodic orbits. The structure of bifurcation diagrams and the characteristics of operation modes at the border collision points are studied too. Finally, PSIM simulation and experimental results testify the validity of theoretical analysis.

A Nonlinear Duopoly with Efficient Production-Capacity Levels

In this paper, we propose an oligopoly model where each firm can produce with linear costs up to its "maximum efficient scale" level and then it incurs quadratic costs for the production exceeding that level. As a consequence, the discrete dynamic model of firms' choices over time is expressed by a piecewise-smooth map. For this model, we show how border collision bifurcations are responsible of the main qualitative changes in the dynamic of the system. In particular, employing specific numerical examples, we explain the basic mechanism of creation/destruction of periodic points and limit cycles through border crossings as the parameters of the model are changed.

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.

Occurrence of multiple attractor bifurcations in the two-dimensional piecewise linear normal form map

Multiple attractor bifurcations occurring in piecewise smooth dynamical systems may lead to potentially damaging situations. In order to avoid these in physical systems, it is necessary to know their conditions of occurrence. Using the piecewise-linear 2D normal form, we investigate which types of multiple attractor bifurcations may occur and where in the parameter space they can be expected. For piecewise smooth maps, multiple attractor bifurcations will be expected to occur if the condition we identified for the piecewise-linear 2D normal form are satisfied in the close neighborhood of the border.

Complex subharmonic oscillation phenomenon of peak current controlled buck converter with current source load

Peak current controlled buck converter with current source load shows the complex phenomenon of fast-scale and slow-scale subharmonic oscillations. Its piecewise smooth switching model and discrete iterative map model are established in this paper. According to its discrete iterative map model, the effects of circuit parameters on the nonlinear dynamic behaviors of buck converter are analyzed by numerical simulation. It is found that attractive regions of fast-scale and slow-scale subharmonic oscillations and dual tori phenomena exist in the bifurcation diagram and Poincaré mapping, respectively. Time-domain simulation waveforms and phase portraits of buck converter are obtained by Runge-Kutta algorithm using the piecewise smooth switching model. The research results indicate that n-type subharmonic oscillation constituted by subharmonic oscillation and frequency-reduced subharmonic oscillation exists in the inductor current, and sine-type subharmonic oscillation constituted by fast scale and low scale exists in the output voltage, respectively. Analysis and simulation results are verified by experimental results.

Border-collision bifurcations in a generalized piecewise linear-power map

In this paper a class of generalized piecewise smooth maps is studied, which is linear at one side and nonlinear with power dependence at the other side. According to the value of the power in the term $x^z$, the bifurcations occurring in this map are classified into five types: $z>1$, $z=1$, $0<z<1$, $z=0$, and $z<0$. We derive the occurrence conditions of border collision bifurcation and smooth fold and flip bifurcation, especially the codimension-2 bifurcation points describing the interaction between border collision bifurcation and smooth bifurcation. The general results are then applied to the specific cases of the power $z$, and different bifurcation scenarios are shown for individual cases, from which the period-adding scenario is found to be general for any power.

Three-Dimensional Cardiac Tissue Image Registration for Analysis of In Vivo Electrical Mapping

A method is presented for registering 3D cardiac tissue images to reference data, for the purpose of analyzing recorded electrical activity. Following left-ventricular in vivo electrical mapping studies in pig hearts, MRI is used to define a reference geometry in the tissue segment around the recording electrodes. The segment is then imaged in 3D using a high-resolution serial imaging microscopy technique. The tissue processing required for this introduces segment-wide distortion. Piecewise-smooth maps are used to correct the tissue distortion and register the 3D images with the reference MRI data. The methods are validated and techniques for identifying the preferred maps are proposed. Recorded electrical activation is shown to map reliably onto cardiac tissue structure using this registration method.