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Considering a piecewise smooth map describing the behavior of a pulse-modulated control system, we discuss border-collision related phenomena. We show that in the parameter space which corresponds to the domain of oscillatory mode a mapping is piecewise linear continuous. It is well known that in piecewise linear maps, classical bifurcations, for example, period doubling, tangent, fold bifurcations become degenerate ("degenerate bifurcations"), combining the properties of both smooth and border-collision bifurcations. We found unusual properties of this map, that consist in the fact that border-collision bifurcations of codimension one, including degenerate ones, occur when a pair of points of a periodic orbit simultaneously collides with two switching manifolds. This paper also discuss bifurcations of chaotic attractors such as merging and expansion ("interior") crises, associated with homoclinic bifurcations of unstable periodic orbits. [ABSTRACT FROM AUTHOR]

Proportional threshold harvesting (PTH) refers to some control rules employed in fishing policies, which specify a biomass level below which no fishing is permitted (the threshold), and a fraction of the surplus above the threshold is removed every year. When these rules are applied to a discrete population model, the resulting map governing the harvesting model is piecewise smooth, so border-collision bifurcations play an essential role in the dynamics. In this paper, we carry out a bifurcation analysis of a PTH model, providing a thorough picture of the 2-parameter bifurcation diagram in the plane (T , q) for a case study. Here, T is the threshold and q is the harvest proportion. Our results explain some numerical bifurcation diagrams in previous work for PTH, and uncover new features of the dynamics with interesting consequences for population management. [ABSTRACT FROM AUTHOR]

We analyze a financial market model with heterogeneous interacting agents where fundamentalists and chartists are considered. We assume that fundamentalists are homogeneous in their trading strategy but heterogeneous in their belief about the asset's fundamental value. On the other hand, we consider that chartists, when they are optimistic become overconfident and they trade more than when they are pessimistic. Consequently, our model dynamics are driven by a one-dimensional piecewise-linear continuous map with three linear branches. We investigate the bifurcation structures in the map's parameter space and describe the endogenous fear and greed market dynamics arising from our asset-pricing model. [ABSTRACT FROM AUTHOR]

Piecewise linear recurrent neural networks (PLRNNs) form the basis of many successful machine learning applications for time series prediction and dynamical systems identification, but rigorous mathematical analysis of their dynamics and properties is lagging behind. Here, we contribute to this topic by investigating the existence of n-cycles (n ≥ 3) and border-collision bifurcations in a class of m-dimensional piecewise linear continuous maps which have the general form of a PLRNN. This is particularly important as for one-dimensional maps the existence of 3-cycles implies chaos. It is shown that these n-cycles collide with the switching boundary in a border-collision bifurcation, and parametric regions for the existence of both stable and unstable n-cycles and border-collision bifurcations will be derived theoretically. We then discuss how our results can be extended and applied to PLRNNs. Finally, numerical simulations demonstrate the implementation of our results and are found to be in good agreement with the theoretical derivations. Our findings thus provide a basis for understanding periodic behavior in PLRNNs, how it emerges in bifurcations, and how it may lead into chaos. [ABSTRACT FROM AUTHOR]

This report presents rigorous analysis of Arnol'd tongues generated by a manifold piecewise linear circuit. The circuit comprises an LC oscillator, a conductive switch that produces the manifold, and a voltage source controlled by the current value through the inductor when the voltage across the capacitor exceeds a threshold. From the circuit dynamics, the Poincaré return map is explicitly derived as a piecewise linear circle map, and the generation of Arnol'd tongues is precisely explained. We attempt to derive the border–collision bifurcation boundaries of the Arnol'd tongues, some of which are obtained explicitly. Furthermore, we experimentally verify these theoretical results. © 2020 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

The recent literature has analyzed binary choices dynamics providing interesting results. Most of these contributions consider interactions within a single group. Nevertheless, in some situations the interaction takes place not only within a single group but also between different groups. In this paper, we investigate the choice dynamics when considering two populations where one serves as a reference group. Considering this influence effect enriches the dynamics. Although the structurally stable resulting dynamics are attracting cycles only, with any positive integer period, the reference group makes the dynamics of the influenced population much more complex. We considered both the possibility that the reference group has the same or the opposite attitude toward the distribution over the choices. We show how the dynamics and the bifurcation structure are modified under the influence of the reference group. Our results illustrate how the propensity to switch choices in the reference groups may, indirectly, affect choices in the first group. [ABSTRACT FROM AUTHOR]

We develop a financial market model with interacting chartists and fundamentalists and chase sellers, the model dynamics is driven by a two-dimensional discontinuous piecewise linear map. Assume that the fixed point on the left side of border is restricted to regular saddle, we provide a more or less complete analytical treatment of the model dynamics by characterizing its possible outcomes in parameter space. The interpretation of structure for basin boundary and chaotic attractor is given by using contact bifurcation resulting from the contact between invariant set and the border. The critical value of occurring boundary crisis is given. In addition, we show that quite different scenarios can trigger real world phenomena such as bull and bear market dynamics and excess volatility. [ABSTRACT FROM AUTHOR]

A model of memristor-based Chua's oscillator is studied. The considered system has infinitely many equilibrium points, which build a line of equilibria. Bifurcational mechanisms of oscillation excitation are explored for different forms of nonlinearity. Hard and soft excitation scenarios have principally different nature. The hard excitation is determined by the memristor piecewise-smooth characteristic and is a result of a border-collision bifurcation. The soft excitation is caused by addition of a smooth nonlinear function and has distinctive features of the supercritical Andronov-Hopf bifurcation. Mechanisms of instability and amplitude limitation are described for both two cases. Numerical modeling and theoretical analysis are combined with experiments on an electronic analog model of the system under study. The issues concerning physical realization of the dynamics of systems with a line of equilibria are considered. The question on whether oscillations in such systems can be classified as the self-sustained oscillations is raised. [ABSTRACT FROM AUTHOR]

Power electronic DC/AC converters (inverters) play an important role in modern power engineering. These systems are also of considerable theoretical interest because their dynamics is influenced by the presence of two vastly different forcing frequencies. As a consequence, inverter systems may be modeled in terms of piecewise smooth maps with an extremely high number of switching manifolds. We have recently shown that models of this type can demonstrate a complicated bifurcation structure associated with the occurrence of border collisions. Considering the example of a PWM H-bridge single-phase inverter, the present paper discusses a number of unusual phenomena that can occur in piecewise smooth maps with a very large number of switching manifolds. We show in particular how smooth (pitchfork and flip) bifurcations may form a macroscopic pattern that stretches across the overall bifurcation structure. We explain the observed bifurcation phenomena, show under which conditions they occur, and describe them quantitatively by means of an analytic approximation. [ABSTRACT FROM AUTHOR]

The goal of this paper is to review some work on agent-based financial market models in which the dynamics is driven by piecewise-linear maps. As we will see, such models allow deep analytical insights into the functioning of financial markets, may give rise to unexpected dynamics effects, allow explaining a number of important stylized facts of financial markets, and offer novel policy recommendations. However, much remains to be done in this rather new research field. We hope that our paper attracts more scientists to this area.

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 naïve 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. [ABSTRACT FROM AUTHOR]

Abstract: The development of switch-mode operated power electronic converter systems has provided a broad range of new effective approaches to the conversion of electric power. In this paper we describe the transitions from regular periodic operation to quasiperiodicity and high-periodic resonance behavior that can be observed in a pulse-width modulated DC/AC converter operating with high feedback gain. We demonstrate the occurrence of two different types of torus birth bifurcations and present a series of phase portraits illustrating the appearance of phase-synchronized quasiperiodicity. Our numerical findings are verified through comparison with an experimental inverter system. The results shed light on the transitions to quasiperiodicity and to various forms of three-frequency dynamics in non-smooth systems. [Copyright &y& Elsevier]

By virtue of their limited size and relatively low costs, multilevel dc–dc converters have come to play an important role in modern industrial power supply systems. When operating in a regime of high corrector gain, such converters can display a variety of new dynamic phenomena associated with the appearance of additional oscillatory modes. Starting in a state where four pairs of stable and unstable period-6 cycles coexist with the basic period-1 cycle, the paper describes a sequence of smooth and nonsmooth bifurcations through which the cycles and their basins of attraction transform as the output voltage is increased. The paper also describes the birth of a multilayered resonance torus through a transverse pitchfork bifurcation of the saddle cycle on an ordinary resonance torus. [ABSTRACT FROM PUBLISHER]

This paper considers the processes of torus formation and reconstruction through smooth and nonsmooth bifurcations in a pulse-width modulated DC/DC converter with multilevel control. When operating in a regime of high corrector gain, converters of this type can generate structures of stable tori embedded one into the other and with their basins of attraction delineated by intervening repelling tori. The paper illustrates the coexistence of three stable tori with different resonance behaviors and shows how reconstruction of these tori takes place across the borders of different dynamical regimes. The paper also demonstrates how pairs of attracting and repelling tori emerge through border-collision torus-birth and border-collision torus-fold bifurcations. [ABSTRACT FROM AUTHOR]

Pulse-modulated converter systems play an important role in modern power electronics. However, by virtue of the complex interplay between ordinary (smooth) and so-called border-collision bifurcations generated by the switching dynamics, the changes in behavior that can occur in multilevel converter systems under varying operational conditions still remain to be explored in full. Considering the dynamics of a three-level dc/dc-converter, we demonstrate a number of new scenarios for the birth or destruction of resonant and ergodic tori. One scenario involves the formation of a doubled-layered torus structure around a stable focus point through three subsequent border-collision fold bifurcations. Another scenario replaces one of the fold bifurcations by a global bifurcation. In both of these scenarios, the basic mode of the converter remains stable while other modes grow up and bifurcate around it. We also illustrate the subcritical birth of both an ergodic and a resonance torus from the basic operational mode. [ABSTRACT FROM AUTHOR]

Abstract: In piecewise-smooth dynamical systems, the regions of existence of a periodic orbit are typically parameter sub-spaces confined by border-collision bifurcations of this orbit. We demonstrate that additionally to the usual border-collision bifurcations occurring at finite points in the state space there exist also border-collision bifurcations occurring at infinity. [Copyright &y& Elsevier]

Abstract: Pulse-modulated converter systems play an important role in modern power electronics. Systems of this type also deserve considerable theoretical interest because of the complex interplay they exhibit between ordinary (smooth) bifurcations and so-called border-collision bifurcations generated by the switching dynamics. Particularly interesting are the unusual transitions to torus dynamics, i.e., to a mode of behavior in which the regular switching dynamics is modulated by another oscillatory mode that may arise through instability in the feedback control. Using the model of a two-level DC/DC converter as an example the paper provides a survey of three new mechanisms of torus bifurcation that can be observed in pulse-modulated control systems. The paper concludes with a discussion of the influence that operation in the torus regimes will have on the efficiency of the converter. [Copyright &y& Elsevier]

The multipliers of a fixed point of a piecewise-smooth, continuous map may change discontinuously as the fixed point crosses a discontinuity under smooth variation of parameters. We study the case when the multipliers "jump" from inside to outside the unit circle, and we assume the map is two-dimensional and piecewise-affine. The resulting dynamics is sometimes similar to the Neimark- Sacker bifurcation of a smooth map in which an attracting periodic or quasiperiodic orbit is created as the fixed point loses stability. However, the bifurcation is often much more complex, with multiple (chaotic) attractors, saddles, and repellors created or destroyed. [ABSTRACT FROM AUTHOR]

It is known that border-collision bifurcations in piecewise-smooth maps can lead to situations where several attractors are created simultaneously in so-called "multiple-attractor" or "multiple-choice" bifurcations. It has been shown that such a situation leads to a fundamental source of uncertainty regarding which attractor the system will follow as a parameter is varied through the bifurcation point. Phenomena of this type have been observed in various physical and engineering systems. We have recently demonstrated that piecewise-smooth systems can exhibit a new type of border-collision bifurcation in which a stable invariant curve, associated with a quasiperiodic or a mode-locked periodic orbit, arises from a fixed point. In this paper we consider a particular variant of the multiple-attractor bifurcation in which a stable periodic orbit arises simultaneously with a closed invariant curve. We also show examples of simultaneously appearing stable periodic orbits and of the simultaneous generation of periodic and chaotic attractors. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus. [Copyright &y& Elsevier]

Abstract: Border-collision bifurcations arise when the periodic trajectory of a piecewise-smooth system under variation of a parameter crosses into a region with different dynamics. Considering a three-dimensional map describing the behavior of a DC/DC power converter, the Letter discusses a new type of border-collision bifurcation that leads to the birth of a “bilayered torus”. This torus consists of the union of two saddle cycles, their unstable manifolds, and a stable focus cycle. When changing the parameters, the bilayered torus transforms through a border-collision bifurcation into a resonance torus containing the stable cycle and a saddle. The Letter also presents scenarios for torus destruction through homoclinic and heteroclinic tangencies. [Copyright &y& Elsevier]

We present a classification of border-collision bifurcations in one-dimensional discontinuous maps depending on the parameters of the piecewise linear approximation in the neighborhood of the point of discontinuity. For each range of parameter values we derive the condition of existence and stability of various periodic orbits and of chaos. This knowledge will help in understanding the bifurcation phenomena in a large number of practical systems which can be modeled by discontinuous maps in discrete domain. [ABSTRACT FROM AUTHOR]

This study investigates the two-frequency quasiperiodic oscillations and Arnol'd tongues generated by two coupled piecewise-constant electric oscillators. The circuit consists of two hysteretic oscillators connected by a capacitor. We take full advantage of the simplicity of this circuit's dynamics, deriving explicit expressions for the infinite number of Arnol'd tongue boundaries, caused by border-collision fold bifurcations. These theoretical results are also verified experimentally. • We investigate two coupled piecewise-constant oscillators. • Piecewise-constant oscillators are the simplest examples of Filippov systems. • Border-collision fold bifurcations occur successively. • The boundaries of infinitely many Arnold tongues are derived explicitly. • The theoretical results are verified experimentally. [ABSTRACT FROM AUTHOR]