Border Collision Bifurcations Research Articles

Can a border collision bifurcation of a chaotic attractor lead to its expansion?

Recently, we reported that a chaotic attractor in a discontinuous one-dimensional map may undergo a so-called exterior border collision bifurcation, which causes additional bands of the attractor to appear. In the present paper, we suppose that the chaotic attractor’s basin boundary contains a chaotic repeller, and discuss a bifurcation pattern consisting of an exterior border collision bifurcation and an expansion bifurcation (interior crisis). In the generic case, where neither the border point the chaotic attractor collides with, nor any of its images belong to the chaotic repeller, the exterior border collision bifurcation is followed by the expansion bifurcation, and the distance between both bifurcations may be arbitrarily small but positive. In the non-generic (codimension-2) case, these bifurcations occur simultaneously, so that a border collision bifurcation of a chaotic attractor leads directly to its expansion.

Unstable dimension variability and heterodimensional cycles in the border-collision normal form.

Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to emphasize the existence of these phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise linear, they are relatively amenable to an exact analysis. We explicitly identify parameter values for heterodimensional cycles and argue that the existence of heterodimensional cycles between two given saddles can be dense in parameter space. We numerically identify key bifurcations associated with unstable dimension variability by studying a one-parameter subfamily that transitions continuously from where periodic solutions are all saddles to where they are all repellers. This is facilitated by fast and accurate computations of periodic solutions; indeed the piecewise-linear form should provide a useful testbed for further study.

A Synopsis of the Noninvertible, Two-Dimensional, Border-Collision Normal Form with Applications to Power Converters

The border-collision normal form is a canonical form for two-dimensional, continuous maps comprised of two affine pieces. In this paper, we provide a guide to the dynamics of this family of maps in the noninvertible case where the two pieces fold onto the same half-plane. Most significantly we identify parameter regimes for the occurrence of key bifurcation structures, such as period-incrementing, period-adding, and robust chaos. We characterize the simplest and most dominant bifurcations and illustrate various dynamical possibilities such as invariant circles, two-dimensional attractors, and several cases of coexisting attractors. We then apply the results to a classic model of a boost converter for adjusting the voltage of direct current. It is known that for one combination of circuit parameters the model exhibits a border-collision bifurcation that mimics supercritical period-doubling and is noninvertible due to the switching mechanism of the converter. We find that over a wide range of parameter values, even though the dynamics created in border-collision bifurcations is in general extremely diverse, the bifurcation in the model can only mimic period-doubling, although it can be subcritical.

Photovoltaic Systems Based on Average Current Mode Control: Dynamical Analysis and Chaos Suppression by Using a Non-Adaptive Feedback Outer Loop Controller

This paper deals with the modeling and theoretical study of an average-current-mode-controlled photovoltaic power conversion chain. It should be noted that current mode control is a superior scheme for controlling DC–DC power electronic converters for photovoltaic applications. Bifurcation diagrams, largest Lyapunov exponents, Floquet theory, and time series are used to study the dynamics of the system. The theoretical results show the existence of subharmonic oscillations and period-1 oscillations in the system. The results of the numerical simulations showed that when the battery voltage at the output of the converter is fixed and ramp amplitude is taken as a control parameter, the photovoltaic power system exhibits the phenomenon of period doubling leading to chaotic dynamics. Furthermore, bifurcation diagrams showed that both the critical value of ramp amplitude for the occurrence of border collision bifurcation and the critical value of ramp amplitude for the occurrence of period-1 in the proposed system increased with the value of the battery terminal voltage. The numerical results are in accordance with the theoretical ones. Finally, an external control based on a non-adaptive controller having a sinusoidal function as a target is applied to the overall system for the suppression of chaotic behavior.

Border-collision bifurcation route to strange nonchaotic attractors in the piecewise linear normal form map

A new route to strange nonchaotic attractors (SNAs) is investigated in a quasiperiodically driven nonsmooth map. It is shown that the smooth quasiperiodic torus becomes nonsmooth (continuous and non-differentiable) due to the border-collision bifurcation of the torus. As the coefficients change, the nonsmooth torus gets gradually fractal, forming strange nonchaotic attractors. It is termed the border-collision bifurcation route to SNAs. A novel feature of this route is that a large number of SNAs are formed, and the parameter area of SNAs makes up about 40 % of the given parameter regions. These SNAs are identified by the largest Lyapunov exponents and the phase sensitivity exponents. They are also characterized by the distribution of finite-time Lyapunov exponents. Unlike other types of SNAs, the distribution of finite-time Lyapunov exponents has its maximum at a relatively small negative Lyapunov exponent, which contributes largely to lead to the abundance of SNAs.

Border collision bifurcations in a piecewise linear duopoly model

We consider a duopoly model characterized by a two-dimensional non-invertible continuous map T given by piecewise linear functions, with several partitions, defining a duopoly game. The structure of the game is such that it has separate second iterate so that its dynamics can be studied via a one-dimensional composite function, that is piecewise linear with multiple partitions in which the definition of the map changes. The number of partitions may change from 2 to 5, depending on the parameters. The dynamics are characterized by degenerate bifurcations and border collision bifurcations, which are typical in maps having kink points. Here the peculiarity is the multiplicity of the partitions, which leads to bifurcations different from those occurring in maps with only one kink point. We show several bifurcations, coexistence of cycles, attracting and superstable, as well chaotic attractors and chaotic repellors, related to the outcome of particular border collision bifurcations.

In-Depth Analysis of Smooth and Nonsmooth Bifurcations for an Open-Loop Boost Converter Feeding Constant Power Loads in Discontinuous Conduction Mode

This paper presents a study of the existence conditions of limit cycles and mechanisms when losing their stability in an open-loop DC–DC boost converter loaded with a constant power load and operating in discontinuous conduction mode. First, the existence conditions are determined. Then, boundaries associated with different kinds of instabilities such as the classical smooth fold and period-doubling bifurcations and nonsmooth border-collision bifurcations such as fold border-collision bifurcations are elucidated. To perform the stability analysis, an implicit 1D map for the system is derived. From this model, it is obtained that smooth period-doubling and fold bifurcations may occur at certain values of the circuit parameters. Some results from numerical simulations performed on the circuit-based switched model are provided showing consistency with the theoretical analysis from the derived model.

Border Collision Bifurcations and Coexisting Attractors in an Economic Bimodal Map

In this paper, we enhance the study of the tâtonnement model with cautious price adjustment introduced by Weddepohl in 1995. Specifically, we investigate the discrete time dynamics of the model when the original assumption of equal values for the maximum rate of increase and decrease of the price is relaxed. As a result, its analytic definition is expressed by a bimodal one-dimensional continuous piecewise smooth map which depends on three parameters. As it happens in general for bimodal maps, it is possible to describe the bifurcation structure in some regions of the parameter space of the model using the skew tent map scenario as a normal form. Nonetheless, we show that some border collision bifurcations which play a fundamental role for the asymptotic behavior of the map essentially pertain to its bimodal shape. Among them we highlight the ones that lead, for some specific parameter values, to the coexistence of two chaotic attractors. Moreover, we identify degenerate border collision bifurcations responsible for the peculiar shapes of the chaotic attractors that distinguish the model.

Multistability and Its Control by the PWM Carrier Signal Modulation in a DC Electric Drive System

Multistability is a universal nonlinear feature for many areas of physics and technology; for most applications, it is an undesired feature leading to uncertainty of the system’s functioning, and therefore, it should be controlled. We deal with the model of a multistable chopper‐fed DC electric drive and study the control of multistability by the periodic modulation of a PWM (pulse‐width modulation) carrier signal. The effect of utilizing sinusoidal modulation has been studied employing brute‐force bifurcation diagrams as well as those obtained by numerical continuation, symbolic analysis of periodic modes, and normal form approximation of border‐collision bifurcation. The result is formulated in terms of the acceptable amplitude of the modulation while the effect of phase shift is taken into account. As an alternative, a modified modulation in the form of the absolute value of the sine wave has been studied; the latter is characterized by a simpler bifurcation structure. Additionally, we propose an implementation of the adaptive control with the objective to drive the system away from a “nonnominal” mode. It has been demonstrated that such a transition should use smooth attenuation of the modulation signal to ensure the desired “nominal” operation of the system.

Revealing bifurcation mechanisms in a 2D nonsmooth map by means of the first return map

In this work, our goal is to show how a properly constructed first return map can help to describe the dynamics of a two-dimensional piecewise smooth map. In particular, to understand some formation principles of the bifurcation structure of its parameter space. The considered map is important from the applied point of view: it models the effects of fraud in a public procurement procedure and allows one to investigate whether honest or dishonest behavior prevails in society. We present the bifurcation structure of the map formed by the periodicity regions associated with attracting cycles of different periods. By means of the first return map, we show that the boundaries of these regions are related to border collision bifurcations typical for nonsmooth maps, as well as standard fold and flip bifurcations. The main economic result is related to the emergence of unexpected changes in qualitative non-compliant behavior over time.

Bifurcations of Sleep Patterns due to Homeostatic and Circadian Variation in a Sleep-Wake Flip-Flop Model

Differential equation-based physiological models of sleep-wake networks describe sleep-wake regulation by simulating the activity of wake- and sleep-promoting neuronal populations and the modulation of these populations by homeostatic and circadian ($\sim24$ h) drives. Here, we consider a sleep-wake flip-flop network model consisting of mutually inhibitory interactions between wake- and sleep-promoting neuronal populations. Motivated by changes in sleep behavior during early childhood as babies transition from napping to non-napping behavior, we vary homeostatic and circadian modulation and analyze effects on resulting sleep-wake patterns. To identify the types and sequences of bifurcations leading to changes in stable sleep-wake patterns in this piecewise-smooth model, we employ multiple mathematical methods, including fast-slow decomposition and numerical computation of circle maps. We find that the average daily number of sleeps exhibits a period adding sequence as the homeostatic time constants are reduced, and that the temporal circadian profile influences the number of observed solutions in the sequence. These solutions emerge through sequences of saddle-node and border collision bifurcations, where the particular sequence depends on parameter values. When the temporal circadian profile is steep, as is observed with long day lengths, some sleep patterns are lost and bistability of other patterns may occur. We analyze a limiting case of the temporal circadian waveform, a circadian hard switch model, to understand this loss of solutions. Generally, our holistic analysis approach provides an alternative analysis method for model systems that defy conventional numerical bifurcation analysis techniques.

Big or small? A new economic geography model with an endogenous switch in the market structure

We present a new economic geography (NEG) model with a linear demand function where firms may change the perception of their relative dimension with respect to the local market. If they perceive themselves as big, they behave as Cournot oligopolists, otherwise they behave as monopolistic competitors. We first compare the pure cases in which only one market form prevails in the two-region economy. Comparing these pure cases of monopolistic and oligopolistic competition only a quantitative difference emerges. Subsequently, we assume that firms switch behavior and start to interact strategically when the number of local firms is below a threshold; in that case, the market form evolves endogenously. Results change substantially. ‘Break’ and ‘sustain’ points are separated as in standard NEG models with an isoelastic demand function leading to a co-existence of equilibria. Stable partial agglomeration and oscillations with small amplitude are possible. The dynamics are described by a 1D piecewise smooth map, which can be continuous (in the pure monopolistic or oligopolistic cases) or discontinuous (when the market structure evolves endogenously). We analyze the bifurcation structure of the parameter space of the map comparing these cases. We show that the continuous maps have rather standard dynamics, while the discontinuous map is characterised by border collision bifurcations of fixed points and cycles, which lead to rich and complex bifurcation structures.

Bifurcations and Chaos in Periodically Forced Coupled Ramp Neurons

Bifurcations and chaos in periodically forced coupled ramp neurons (rectified linear units) are examined as one of minimal chaotic neural networks. The first ramp neuron connects to the second ramp neuron excitatorily while the second ramp neuron connects to the first neuron inhibitorily. When the first ramp neuron is forced by a rectangular wave, a stable periodic solution is generated. The system is piecewise linear and its periodic solutions are obtained by connecting local solutions in linear domains at borders and solving transcendental equations. Their Poincaré maps and Jacobian matrices are also derived rigorously, by which the bifurcations of the periodic solutions are calculated. The periodic solution undergoes period-doubling bifurcations, which leads to the generation of a chaotic attractor. A period-three solution is also generated through saddle-node bifurcations, which is characteristic of a chaotic system. A one-dimensional phase return map is derived and it is shown that the bifurcations of its fixed points agree with the bifurcations of the multiple periodic solutions of the original forced system. Further, qualitatively the same bifurcations and chaos occur when the first ramp neuron is replaced by a step neuron, in which a chaotic attractor is also generated through a border collision bifurcation. The linear response and the dead zone of the ramp neurons are essential for the emergence of chaos.

Border collision bifurcation of a resonant closed invariant curve.

This paper contributes to studying the bifurcations of closed invariant curves in piecewise-smooth maps. Specifically, we discuss a border collision bifurcation of a repelling resonant closed invariant curve (a repelling saddle-node connection) colliding with the border by a point of the repelling cycle. As a result, this cycle becomes attracting and the curve is destroyed, while a new repelling closed invariant curve appears (not in a neighborhood of the previously existing invariant curve), being associated with quasiperiodic dynamics. This leads to a global restructuring of the phase portrait since both curves mentioned above belong to basin boundaries of coexisting attractors.

Dynamical Analysis of Border Collision Bifurcation in One-Cycle Controlled Single-Inductor Dual-Output Boost DC-DC Converters

In this paper, the evolution of three types of border collision bifurcations are investigated thoroughly for a high-order single-inductor double-output DC-DC converter with one-cycle control (OCC). Multi-piecewise inductor current with more than one border and cross-coupling effect between different output ports are prominent features of the system, resulting in unexpected period incrementing cascade and transition from periodicity to chaos as a parameter is varied. A high-order nonlinear iterative map model is proposed, which reveals the underlying mechanisms of those complicated nonstandard and unsmooth bifurcations in the multiple-output system. Additionally, theoretical analysis shows that those nonstandard bifurcations arise from the boundary collisions, and more specifically, a fixed point suddenly appears in the borderline between various regions. Besides eigenvalue loci changes abruptly at the bifurcation point, the characteristic of border collision bifurcation is the topology sequence changes. Furthermore, the parameter boundaries are given to provide useful guidance for design. Finally, PSpice circuit experiments are performed to verify the above theoretical and numerical results.

Appearance of closed invariant curves in a piecewise Cournot model with advertising

In the present paper we investigate two cases which can explain what happens when the Cournot equilibrium of a duopoly model loses stability through a Neimark-Sacker bifurcation of subcritical type. This kind of bifurcation involves complex dynamics which lead to the appearance of closed invariant curves. We analyze a Cournot model where competitors hold different plants and compete on advertising quantities. The model is described by a two-dimensional piecewise map in discrete time. Making use of analytical results and numerical simulations, we show that the appearance/disapearance of closed invariant curves is directly related to two different mechanisms, namely homoclinic bifurcations and border collision bifurcations.

Bifurcation analysis of stick–slip vibration in a 2-DOF nonlinear dynamical system with dry friction

This paper describes a new approach to study the nonlinear dynamical behaviors of a belt-driven two-coupled friction oscillator. First, using Buckingham’s π theorem, several dimensionless and physically meaningful parameters are derived. Their relationships with the steady state responses of the system are revealed through numerical simulations. Then a new mapping tool named as sticking phase plot is developed to record the transition process of stick–slip motion. It intuitively shows how bifurcation occurs in sticking region and transforms the dynamical responses. The bifurcation diagrams of the system are obtained by sweeping the dimensionless parameters. Some extraordinary bifurcation phenomena are observed in this system. By using sticking phase plot, it is found that the bifurcations can be broadly divided into four main categories: the Border-Collision Bifurcation, the Grazing-Sliding Bifurcation, the Multi-Sliding Bifurcation, and the Fixed-Point Bifurcation. New nonlinear phenomena are found herein, including chaos caused by the Border-Collision Bifurcation and the Sliding Bifurcations. The local and the global changes caused by the Fixed-Point bifurcation are also observed. In numerical simulation, we adopted the LVIM method proposed by the authors, which can be used to integrate non-smooth systems very accurately and efficiently.

Border Collision Bifurcations and Chaos in a Ring of Three Unidirectionally Coupled Nonmonotonic Piecewise Constant Neurons

The bifurcations and chaos in a ring of three nonmonotonic piecewise constant neurons with unidirectional inhibitory coupling are examined. Periodic solutions in the system undergo border collision bifurcations, which are characteristic of piecewise smooth systems. Conditions for the bifurcations of periodic solutions are derived analytically by using a piecewise constant function as the output function of neurons. Multiple periodic solutions are generated through border collision bifurcations and saddle-node bifurcations. Chaotic attractors are generated through border collision bifurcations and degenerate period-doubling bifurcations. The existence of multiple periodic solutions and chaotic attractors is proven to be common in rings of unidirectionally coupled nonmonotonic neurons.

Border-Collision Bifurcation Route to Strange Nonchaotic Attractors in the Piecewise Linear Normal Form Map

Border-Collision Bifurcation Route to Strange Nonchaotic Attractors in the Piecewise Linear Normal Form Map

Analyzing a macroprudential instrument during the COVID-19 pandemic using border collision bifurcation

Rect@ son las iniciales de Revista Electrónica de Comunicaciones y Trabajos de ASEPUMA. Se trata de una iniciativa de la Asociación de Profesores Universitarios de Matemáticas aplicadas a la Economía y la Empresa (ASEPUMA) con el fin de difundir los trabajos que realicen sus asociados, así como otros investigadores, en el campo de las matemáticas aplicadas a la economía y la empresa