Border Collision Bifurcations Research Articles

Bifurcation analysis of a piecewise-smooth Ricker map with proportional threshold harvesting

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 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.

Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo border collision bifurcations, leading to a sudden change in their structure. We describe two types of such border collision bifurcations and explain the mechanisms causing the changes in the geometrical structure of the attractors, in particular, in the number of their bands (connected components).

Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

Multistability Regions and Related Basins of Multiattractors in Piecewise Linear Discontinuous Map from Financial Markets

By adding trend followers, we extend the model given by Tramontana et al. from one-dimensional ([Formula: see text]D) piecewise linear discontinuous (PWLD) map to a new 2D PWLD map. Using this map in financial markets, we describe the bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: border collision bifurcations; Poincaré equator collision bifurcations; degenerate flip bifurcations in both supercritical and subcritical cases. We investigate the multistability regions in the parameter plane and related basins of multiattractors to uncover the reason for the unpredictability of the internal law of price fluctuations in financial market.

Dynamics of a business cycle model with two types of governmental expenditures: the role of border collision bifurcations

Dynamics of a business cycle model with two types of governmental expenditures: the role of border collision bifurcations

Bifurcation phenomena in the peristaltic transport of non-Newtonian fluid with heat and mass transfer effects

An in-depth bifurcation analysis is carried out for the peristaltic transport of non-Newtonian fluid with heat and mass transfer through an axisymmetric channel. Based on the perturbation technique, analytical solutions for flow rate and stream function are presented. This function and its velocity fields build a nonlinear dynamic system in two spatial dimensions. We are then interested in identifying the global and local bifurcation of the invariant curves in which the recurrent fluid dynamics close to the stagnation points change qualitatively. For this purpose, a geometric description, analytical expressions and the development of a computational algorithm are provided to recognize the multiplicity of admissible/virtual stagnation points. In a variety of physical parameters, the analysis highlights the presence of several types of nonlinear phenomena, such as infinite/finite heteroclinic and homoclinic orbits, saddle-node and border-collision bifurcations. These results guide the streamline patterns for capturing novel complex behaviors such as multiple trapping phenomena and critical transition to distinguish between different flow regions.

Chaotic attractors from border-collision bifurcations: stable border fixed points and determinant-based Lyapunov exponent bounds

The collision of a fixed point with a switching manifold (or border) in a piecewise-smooth map can create many different types of invariant sets. This paper explores two techniques that, combined, establish a chaotic attractor is created in a border-collision bifurcation in $\mathbb{R}^d$ $(d \ge 1)$. First, asymptotic stability of the fixed point at the bifurcation is characterised and shown to imply a local attractor is created. Second, a lower bound on the maximal Lyapunov exponent is obtained from the determinants of the one-sided Jacobian matrices associated with the fixed point. Special care is taken to accommodate points whose forward orbits intersect the switching manifold as such intersections can have a stabilising effect. The results are applied to the two-dimensional border-collision normal form focusing on parameter values for which the map is piecewise area-expanding.

On the significance of borders: the emergence of endogenous dynamics

We propose a prototype model of market dynamics in which all functional relationships are linear. We take into account three borders, defined by linear functions, that are intrinsic to the economic reasoning: non-negativity of prices; downward rigidity of capacity (depreciation); and a capacity constraint for the production decision. Given the linear specification, the borders are the only source for the emerging of cyclical and more complex dynamics. In particular, we discuss centre bifurcations, border collision bifurcations and degenerate flip bifurcations—dynamic phenomena the occurrence of which are intimately related to the existence of borders.

Simulation and Analysis of Complex Behaviors in a DC-DC Buck Converter

A new study of the Buck converter controlled by a smooth signal -changing the widely used non-smooth ramp signal by a smooth sine signal- operating in continuous conduction mode (CCM) is carried out in this paper, focusing on the analysis of bifurcation diagrams of relevant parameters that govern the system. Unexpectedly and contrary to what one would intuitively contemplate, new phenomena or complex dynamics of the non-smooth nature appear. Mainly border collisions and other non-smooth bifurcations are the main causes of the chaotic attractors and complex behaviors' emergence, even though the signal that controls the system is smooth. More importantly, a cascade of border collision bifurcations was discovered in the Buck converter controlled by a sine waveform, a kind of non-sooth bifurcation only reported in specific type of power inverters

Analysis of a piecewise linear aeroelastic system with and without tuned vibration absorber

The dynamics of a two-degrees-of-freedom (pitch–plunge) aeroelastic system is investigated. The aerodynamic force is modeled as a piecewise linear function of the effective angle of attack. Conditions for admissible (existing) and virtual equilibria are determined. The stability and bifurcations of equilibria are analyzed. We find saddle-node, border collision and rapid bifurcations. The analysis shows that the pitch–plunge model with a simple piecewise linear approximation of the aerodynamic force can reproduce the transition from divergence to the complex aeroelastic phenomenon of stall flutter. A linear tuned vibration absorber is applied to increase stall flutter wind speed and eliminate limit cycle oscillations. The effect of the absorber parameters on the stability of equilibria is investigated using the Liénard–Chipart criterion. We find that with the vibration absorber the onset of the rapid bifurcation can be shifted to higher wind speed or the oscillations can be eliminated altogether.

On the Price Dynamics of a Two-Dimensional Financial Market Model with Entry Levels

This paper aims to extend the model developed by Tramontana et al. By adding trend followers who pay attention to the most recent observed price trend, we formulate a financial market model driven by a new two-dimensional discontinuous piecewise linear (PWL) map with three branches. The dynamic behavior of the mapping system is studied in two cases according to different trend followers’ expectation of the stock price. The existence and stability conditions of periodic attractors and other bounded attractors are derived by using qualitative and quantitative methods, theoretical analysis, and numerical simulation. When trend followers are neutral on the stock market, we present that the basin of locally attracting fixed points can be determined by the preimages of two borderlines. We also prove that one of its surprising features is that model simulations may appear to be chaotic, although only regular dynamics can emerge. When trend followers are bullish or bearish on the stock market, we present the existence conditions of attracting coexistence fixed point, globally and locally attracting fixed point, and periodic and other bounded attractors. The transversal homoclinic theory of flip saddle periodic point is applied to prove the existence of chaotic attractor. We also give the calculation methods of border collision bifurcation (BCB) curves. This paper advances our knowledge of discontinuous PWL systems and reveals the endogenous evolution of bubbles and crashes and excessive volatility in financial markets from a new perspective with new methods.

Codimension-two border collision bifurcation in a two-class growth model with optimal saving and switch in behavior

We consider a two-class growth model with optimal saving and switch in behavior. The dynamics of this model is described by a two-dimensional (2D) discontinuous map. We obtain stability conditions of the border and interior fixed points (known as Solow and Pasinetti equilibria, respectively) and investigate bifurcation structures observed in the parameter space of this map, associated with its attracting cycles and chaotic attractors. In particular, we show that on the x-axis, which is invariant, the map is reduced to a 1D piecewise increasing discontinuous map, and prove the existence of a corresponding period adding bifurcation structure issuing from a codimension-two border collision bifurcation point. Then, we describe how this structure evolves when the related attracting cycles on the x-axis lose their transverse stability via a transcritical bifurcation and the corresponding interior cycles appear. In particular, we show that the observed bifurcation structure, being associated with the 2D discontinuous map, is characterized by multistability, that is impossible in the case of a standard period adding bifurcation structure.

Existence of n-cycles and border-collision bifurcations in piecewise-linear continuous maps with applications to recurrent neural networks

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 (nge 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.

Dynamics and bifurcations of a family of piecewise smooth maps arising in population models with threshold harvesting.

We study a discrete-time model for a population subject to harvesting. A maximum annual catch H is fixed, but a minimum biomass level T must remain after harvesting. This leads to a mathematical model governed by a continuous piecewise smooth map, whose dynamics depend on two relevant parameters H and T. We combine analytical and numerical results to provide a comprehensive overview of the dynamics with special attention to discontinuity-induced (border-collision) bifurcations. We also discuss our findings in the context of harvest control rules.

Chaos, border collisions and stylized empirical facts in an asset pricing model with heterogeneous agents

An asset pricing model with chartists, fundamentalists and trend followers is considered. A market maker adjusts the asset price in the direction of the excess demand at the end of each trading session. An exogenously given fundamental price discriminates between a bull market and a bear market. The buying and selling orders of traders change moving from a bull market to a bear market. Their asymmetric propensity to trade leads to a discontinuity in the model, with its deterministic skeleton given by a two-dimensional piecewise linear dynamical system in discrete time. Multiple attractors, such as a stable fixed point and one or more attracting cycles or cycles and chaotic attractors, appear through border collision bifurcations. The multi-stability regions are underlined by means of two-dimensional bifurcation diagrams, where the border collision bifurcation curves are detected in analytic form at least for basic cycles with symbolic sequences $${\hbox {LR}}^{n}$$ and $${\hbox {RL}}^{n}$$ . A statistical analysis of the simulated time series of the asset returns, generated by perturbing the deterministic dynamics with a random walk process, indicates that this is one of the simplest asset pricing models which are able to replicate stylized empirical facts, such as excess volatility, fat tails and volatility clustering.

Rigorous analysis of Arnol'd tongues from a manifold piecewise linear circuit

AbstractThis 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.

Bifurcation Sequences and Multistability in a Two-Dimensional Piecewise Linear Map

Bifurcation mechanisms in piecewise linear or piecewise smooth maps are quite different with respect to those occurring in smooth maps, due to the role played by the borders. In this work, we describe bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: (A) fold border collision bifurcations, (B) degenerate flip bifurcations, supercritical and subcritical, (C) degenerate transcritical bifurcations and (D) supercritical center bifurcations. Each of these is characterized by a particular dynamic behavior, and may be related to attracting or repelling cycles. We consider different bifurcation routes, showing the interplay between all these kinds of bifurcations, and their role in the phase plane in determining attracting sets and basins of attraction.

Particle swarm optimization-based strategy for detecting border-collision bifurcation points in piecewise smooth maps

A border-collision bifurcation point detection strategy based on particle swarm optimization (PSO) is proposed in this paper, and it facilitates a bifurcation point detection in piecewise smooth maps. This method is tested via detection experiments of border-collision bifurcation points in two popular piecewise smooth maps with one or more borders. The algorithm design and analysis part shows that the proposed algorithm is fairly simple, easily understandable, and easily implementable. The simulation results show that it accurately detects the border-collision parameters of the target period regardless of its stability, but requires no careful initialization or gradient information of the system.

Unfolding Codimension-Two Subsumed Homoclinic Connections in Two-Dimensional Piecewise-Linear Maps

For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such “subsumed” homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy [Formula: see text], in a two-parameter unfolding there exists an infinite sequence of roughly triangular regions within which the map has a stable single-round periodic solution. The result applies to both discontinuous and continuous maps, although these cases admit different characterizations for the border-collision bifurcations that correspond to boundaries of the regions. The result is illustrated with a discontinuous map of Mira and the two-dimensional border-collision normal form.

Hyperchaos in 3-D piecewise smooth maps

Abstract In this paper, we show various ways of the occurrence of a hyperchaotic orbit in 3D piecewise linear normal form maps. We show that hyperchaotic orbit can be born from a periodic orbit or a quasiperiodic orbit in various ways like-(a) a direct transition to a hyperchaotic orbit from a periodic orbit or a from a quasiperiodic orbit through border collision bifurcation; (b) a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic and chaotic orbit; (c) a transition from a mode-locked periodic orbit to a hyperchaotic orbit via higher dimensional torus. We also show bifurcations where a hyperchaotic orbit bifurcates to a different hyperchaotic orbit or a three-piece hyperchaotic orbit. We further show period increment with the coexistence of hyperchaotic attractors. Moreover, we numerically calculate the existence region of a hyperchaotic orbit in the parameter space region.