Border Collision Bifurcations Research Articles

Deep learning for prediction and classifying the dynamical behaviour of piecewise-smooth maps

This paper explores novel ways of predicting and classifying the dynamics of piecewise smooth maps using various deep learning models. Moreover, we have used machine learning models such as Decision Tree Classifier, Logistic Regression, K-Nearest Neighbor, Random Forest, and Support Vector Machine for predicting the border collision bifurcation in the 1D normal form map and the 1D tent map. The decision tree classifier best predicts the border collision bifurcation for the 1D normal form map, the random forest, and the 1D tent map. This study introduces a novel application of deep learning models to cobweb diagrams and phase portraits, which provides a new perspective for classifying regular and chaotic behaviour. Further, we classified the regular and chaotic behaviour of the 1D tent map and the 2D Lozi map using deep learning models like Convolutional Neural Network (CNN), ResNet50, and ConvLSTM via cobweb diagram and phase portraits, where CNN exhibits better performance than other models. We also classified the chaotic and hyperchaotic behaviour of the 3D piecewise smooth map using deep learning models such as the Feed Forward Neural Network (FNN), Long Short-Term Memory (LSTM), and Recurrent Neural Network (RNN). We have shown that LSTM performs best for classifying chaotic and hyperchaotic behaviour. Additionally, LSTM outperforms other models in accuracy and computational efficiency, making it highly effective for real-time analysis. Finally, deep learning models such as Long Short Term Memory (LSTM) and Recurrent Neural Network (RNN) are used for reconstructing the two-parameter bifurcation charts of 2D normal form map, in which LSTM is more precise than RNN in reconstructing the two-parametric charts.

Spectral Signatures of Bifurcations

One way to characterize an orbit of a dynamical system is through its frequency content. Using a “spectral bifurcation diagram”, it has been shown earlier how the frequency content changes when a system undergoes a period-doubling cascade. In this paper, we extend the scope of this technique to obtain newer insights into various bifurcations like pitchfork bifurcation, border-collision bifurcation, period-adding cascade, and various torus bifurcations. We show that applying this method can enrich our understanding of bifurcations by providing vital information about generating or annihilating frequency components in a bifurcation.

Discrete model and non‐linear characteristics analysis of magnetic coupled resonant wireless power transfer system with constant power load

AbstractMagnetically coupled resonant wireless power transfer (MCR‐WPT) system with constant power load (CPL), finds extensive applications in military, industrial, and medical treatment. However, this system can easily exhibit complex non‐linear behaviors under certain parameter conditions due to the influences of constant power loads, switching devices, and feedback controllers. These behaviors limit the effectiveness of controllers and reduce the efficiency and stability of the system. It is necessary to study the non‐linear characteristics of the MCR‐WPT system with CPL. The stroboscopic discrete mapping of the MCR‐WPT system with CPL is established. Then the system's non‐linear dynamics are analyzed theoretically using a bifurcation diagram, the maximal Floquet multipliers, and the maximal Lyapunov exponent spectrum obtained from the proposed discrete mapping. The results show that the MCR‐WPT system with CPL will exhibit rich non‐linear dynamics with the variation of the power of CPL, such as cyclic fold bifurcation, Neimark–Scaker bifurcation, border collision bifurcations, chaos, etc. The excellent alignment of experimental and theoretical outcomes in corresponding states confirms the accuracy of the proposed discrete mapping and the nonlinear analysis of the system. The results of this study can provide a reference for selecting parameters for an actual MCR‐WPT system with CPL.

Sigmoid function model of parallel-connected DC-DC converters and analysis of their dynamic characteristics.

Because of the manufacturing variations of circuit elements and the effects of parasitic parameters, the switching elements of parallel-connected modules cannot be synchronously switched. Consequently, the characteristics of these converters, which are multi-mode, high-order systems, cannot be comprehensively described using reduced-order models based on the symmetry of circuit topology and parameters. It is also difficult to fully describe the characteristics of these systems using discrete mapping models and averaged models. Therefore, we developed a smooth, sigmoid function-based continuous-time model for systems comprising parallel-connected buck converters with average current-sharing control. The proposed model provides a unified description of the continuous and discontinuous conduction modes of the system. The criteria for ensuring the stability of the system were derived based on the Floquet theory. The nonlinear dynamic behavior of the system and the effects of the key parameters on the stability of the system were investigated. The results showed that the system may undergo period doubling bifurcation, border collision bifurcation, and Neimark-Sacker bifurcation as the system parameters varied. The theoretical analysis and simulation results were verified experimentally. Our results provide a basis for the configuration of controller parameters.

Bifurcation structures of a two-dimensional piecewise linear discontinuous map: analysis of a cobweb model with regime-switching expectations

We consider the bifurcations occurring in a two-dimensional piecewise-linear discontinuous map that describes the dynamics of a cobweb model in which firms rely on a regime-switching expectation rule. In three different partitions of the phase plane, separated by two discontinuity lines, the map is defined by linear functions with the same Jacobian matrix, having two real eigenvalues, one of which is negative and one equal to 0. This leads to asymptotic dynamics that can belong to two or three critical lines. We show that when the basic fixed point is attracting, it may coexist with at most three attracting cycles. We have determined their existence regions, in the two-dimensional parameter plane, bounded by border collision bifurcation curves. At parameter values for which the basic fixed point is repelling, chaotic attractors may exist - either one that is symmetric with respect to the basic fixed point, or, if not symmetric, the symmetric one also exists. The homoclinic bifurcations of repelling cycles leading to the merging of chaotic attractors are commented by using the first return map on a suitable line. Moreover, four different kinds of homoclinic bifurcations of a saddle 2-cycle, leading to divergence of the generic trajectory, are determined.

Bifurcations and complex dynamics in a banking duopoly model with macroprudential policy

We consider a banking duopoly model with a macroprudential policy in Indonesia called loan-to-deposit ratio-based reserve requirement (LDR-RR). The objective of the policy is to control the banking loans growth using a LDR-based penalty scheme that requires banks to save more money in the central bank to maintain their liquidity ratio. Following recent studies on banking models, we analyze a piecewise discrete-time model with two banks. We assume that the dynamics of the deposits follows the discrete logistic growth. Moreover, our model has two borders, hence we examine the resulting border-collision bifurcations. From the local stability analysis we find that, according to the parameter values, only the border-collision bifurcation or the flip bifurcation occurs. Finally, we perform several numerical simulations to confirm the stability analysis’ results. Our discrete dynamical system offers various possibilities of development for future research perspectives.

On Forced Oscillations in a Relay System with Hysteresis

This paper discusses the phenomenon associated with the forced synchronization (“entrainment of a self-sustained oscillator by an external force”) in a relay system with hysteresis, which manifests itself in the occurrence of periodic motions close to the rhythmic activity of neurons, when packets of fast oscillations are interspersed with intervals of the slow dynamics. To study this phenomenon, we introduce a circle mapping, which, depending on the parameters, can be a circle diffeomorphism or discontinuous map (“gap map”). In both cases, this mapping demonstrates the so-called period-adding bifurcation structure. It is demonstrated that packets number of fast oscillations in the period of periodic motion is determined by the rotation number, and the length of the intervals between the packets may be found of the boundaries of the absorbing interval. The change in the number of pulses in the packet occurs through the border-collision bifurcation.

The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps

Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period-p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence, where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold, where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on Rn to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle.

Inclusion of higher-order terms in the border-collision normal form: Persistence of chaos and applications to power converters

The dynamics near a border-collision bifurcation are approximated to leading order by a continuous, piecewise-linear map. The purpose of this paper is to consider the higher-order terms that are neglected when forming this approximation. For two-dimensional maps we establish conditions under which a chaotic attractor created in a border-collision bifurcation persists for an open interval of parameters beyond the bifurcation. We apply the results to a prototypical power converter model to prove the model exhibits robust chaos.

On Border-Collision Bifurcations in a Pulse System

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.

Study the bifurcations of a 2DoF mechanical impacting system

Impacting mechanical systems with suitable parameter settings exhibit a large amplitude chaotic oscillation close to the grazing with the impacting surface. The cause behind this uncertainty is the square root singularity and the occurrence of dangerous border collision bifurcation. In the case of one-degree-of-freedom mechanical systems, it has already been shown that this phenomenon occurs under certain conditions. This paper proposes the same uncertainty of a two-degree freedom mechanical impacting system under specific requirements. This paper shows that the phenomena earlier reported in the case of one-degree-of-freedom mechanical systems (like narrow band chaos, finger-shaped attractor, etc.) also occur in the two-degrees-of-freedom mechanical impacting system. We have numerically predicted that the narrowband chaos ensues under specific parameter settings. We have also shown that narrowband chaos can be avoided under some parameter settings. At last, we demonstrate the numerical predictions experimentally by constructing an equivalent electronic circuit of the mechanical rig.

1d piecewise smooth map: exploring a model of investment dynamics under financial frictions with three types of investment projects

UDC 517.9 We consider a 1D continuous piecewise smooth map, which depends on seven parameters. Depending on the values of parameters, it may have up to six branches. This map was proposed by Matsuyama [Theor. Econ., <strong>8</strong>, 623–651 (2013); Section 5]. It describes the macroeconomic dynamics of investment and credit fluctuations in which three types of investment projects compete in the financial market. We introduce a partitioning of the parameter space according to different branch configurations of the map and illustrate this partitioning for a specific parameter setting. Then we present an example of the bifurcation structure in a parameter plane, which includes periodicity regions related to superstable cycles. Several bifurcation curves are obtained analytically, in particular, the border-collision bifurcation curves of fixed points. We show that the intersection point of two curves of this kind is an organizing center from which infinitely many other bifurcation curves are originated.

On bifurcations of chaotic attractors in a pulse width modulated control system

This paper discusses bifurcational phenomena in a control system with pulse-width modulation of the first kind. We show that the transition from a regular dynamics to chaos occurs in a sequence of classical supercritical period doubling and border collision bifurcations. As a parameter is varied, one can observe a cascade of doubling of the cyclic chaotic intervals, which are associated with homoclinic bifurcations of unstable periodic orbits. Such transition are also refereed as merging bifurcation (known also as merging crisis). At the bifurcation point, the unstable periodic orbit collides with some of the boundaries of a chaotic attractor and as a result, the periodic orbit becomes a homoclinic. This condition we use for obtain equations for bifurcation boundaries in the form of an explicit dependence on the parameters. This allow us to determine the regions of stability for periodic orbits and domains of the existence of four-, two- and one-band chaotic attractors in the parameter plane.

Border-collision bifurcation in the Logistic map with state-dependent impulsive forces

Border-collision bifurcation in the Logistic map with state-dependent impulsive forces

Regular and chaotic dynamics in a 2D discontinuous financial market model with heterogeneous traders

We develop a financial market model where three types of traders operate simultaneously: fundamentalists and chartists of two types, namely, trend followers and contrarians. The dynamics of this model is described by a two-dimensional discontinuous map defined by two linear functions, where one acts in the partition between two (parallel) discontinuity lines and the other one acts outside this partition. Our analysis shows that despite the linearity of the map components, its dynamics can be quite complex, with various, possibly coexisting attracting cycles and chaotic attractors. As a first step towards the understanding how the overall bifurcation structure observed in the parameter space of the map is organized, we obtain analytically the boundaries of periodicity regions related to the simplest attracting period-n cycles, n≥3, with one point in the middle partition and n−1 points outside it. These boundaries can be related to border-collision bifurcations (when a point of the cycle collides with a discontinuity line) as well as to degenerate bifurcations (associated with eigenvalues on the unit circle). We show also some elements of period-adding and period-incrementing bifurcation structures for which the cycles mentioned above are basic. From an economic point of view, our study confirms that a fairly simple financial market model with heterogeneous agents is able to produce complicated boom-bust dynamics typical of real financial markets.

Structurally Unstable Synchronization and Border-Collision Bifurcations in the Two-Coupled Izhikevich Neuron Model

This study investigates a structurally unstable synchronization phenomenon observed in the two-coupled Izhikevich neuron model. As the result of varying the system parameter in the region of parameter space close to where the unstable synchronization is observed, we find significant changes in the stability of its periodic motion. We derive a discrete-time dynamical system that is equivalent to the original model and reveal that the unstable synchronization in the continuous-time dynamical system is equivalent to border-collision bifurcations in the corresponding discrete-time system. Furthermore, we propose an objective function that can be used to obtain the parameter set at which the border-collision bifurcation occurs. The proposed objective function is numerically differentiable and can be solved using Newton’s method. We numerically generate a bifurcation diagram in the parameter plane, including the border-collision bifurcation sets. In the diagram, the border-collision bifurcation sets show a novel bifurcation structure that resembles the “strike-slip fault” observed in geology. This structure implies that, before and after the border-collision bifurcation occurs, the stability of the periodic point discontinuously changes in some cases but maintains in other cases. In addition, we demonstrate that a border-collision bifurcation set successively branches at distinct points. This behavior results in a tree-like structure being observed in the border-collision bifurcation diagram; we refer to this structure as a border-collision bifurcation tree. We observe that a periodic point disappears at the border-collision bifurcation in the discrete-time dynamical system and is simultaneously replaced by another periodic point; this phenomenon corresponds to a change in the firing order in the continuous-time dynamical system.

A robust route to randomness in a simple Cournot duopoly game where ambiguity aversion meets constant expectations

In this paper we investigate the dynamics of a duopoly game with ambiguity aversion regarding uncertainty in demand and constant expectations concerning competitor production. The focus is on an asymmetric Cournot game where players engage in robust optimization and have different beliefs about the possible realizations of the random parameters of the price function. The players’ ambiguity aversion introduces multiple equilibria and instability that otherwise would not be present. The investigation of the global dynamics of the game reveals the emergence, through border-collision bifurcations, of periodic and chaotic dynamics.

Dynamic Investigations of an Endogenous Business Cycle Model with Heterogeneous Agents

AbstractWe introduce agents’ heterogeneity into a model of endogenous business cycles, in which agents can invest either in ‘good’ projects that contribute to future capital formation, or in ‘bad’ projects without that property. The resulting map involves three distinct regimes, two of which we linearize. Using theoretical results on piecewise linear systems and on border collision bifurcations, we are able to provide a thourough analysis of the dynamics.

Dynamic tipping and cyclic folds, in a one-dimensional non-smooth dynamical system linked to climate models

We study the behaviour at tipping points close to (smoothed) non-smooth fold bifurcations in one-dimensional oscillatory forced systems with slow parameter drift. The focus is the non-smooth fold in the Stommel 2-box, and related climate models, which are piecewise-smooth continuous dynamical systems, modelling thermohaline circulation. These exhibit non-smooth fold bifurcations which arise when a saddle-point and a focus meet at a border collision bifurcation. By using techniques from the theory of non-smooth dynamical systems we are able to provide precise estimates for the general tipping behaviour close to the non-smooth fold as parameters vary. These are different from the usual tipping point estimates at a saddle–node bifurcation, with advanced tipping apparent in the non-smooth case when compared to the behaviour near smoothed approximations to this fold. We also see rapid, and non-monotone, transitions in the tipping points for oscillatory forced systems close to both non-smooth folds and saddle–node bifurcations due to the effects of both phase changes and non-smoothness. These variations can have implications for the prediction of tipping in climate systems, particularly in close proximity to a non-smooth fold.

Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors

In diverse physical systems stable oscillatory solutions devolve into more complicated solutions through border-collision bifurcations. Mathematically these occur when a stable fixed point of a piecewise-smooth map collides with a switching manifold as parameters are varied. The purpose of this paper is to highlight the extreme complexity possible in the subsequent dynamics. By perturbing instances of the n-dimensional border-collision normal form for which the n th iterate is a direct product of chaotic skew tent maps, it is shown that many chaotic attractors can arise. Burnside's lemma is used to count the attractors; chaoticity is proved by demonstrating that some iterate of the map is piecewise-expanding. The resulting transition from a stable fixed point to many coexisting chaotic attractors occurs throughout open subsets of parameter space and is not destroyed by higher order terms, hence can be expected to occur generically in mathematical models.