Piecewise Linear Dynamical Systems Research Articles

Optimal switching boundary control for two-dimensional piecewise linear dynamical systems

This paper deals with the optimal control design for a family of two-dimensional piecewise linear systems having a switching boundary defined by a straight line and vector fields with regular equilibrium points. Such a family is derived from discontinuous control systems characterized by exhibiting sliding motion and a single pseudo-equilibrium point at the switching boundary, which is the desired operating point of the system in the control design. The initial objective is to obtain conditions on the slope of the switching boundary so that the pseudo-equilibrium is asymptotically stable, at least locally. The second and main objective is, based on the pseudo-equilibrium stability result, to find the optimal slope for the switching boundary in order to minimize the transient response time. A numerical method for obtaining the optimal slope is presented, and the effects caused by uncertainties in the parameters are discussed. The used approach explores an innovative way of determining the optimal slope of the switching boundary based on finite-time stability conditions. The results obtained are applied to a power electronic system that describes the dynamics of a Buck Converter under a sliding mode control strategy.

Variational Inference and Learning of Piecewise Linear Dynamical Systems.

Modeling the temporal behavior of data is of primordial importance in many scientific and engineering fields. Baseline methods assume that both the dynamic and observation equations follow linear-Gaussian models. However, there are many real-world processes that cannot be characterized by a single linear behavior. Alternatively, it is possible to consider a piecewise linear model which, combined with a switching mechanism, is well suited when several modes of behavior are needed. Nevertheless, switching dynamical systems are intractable because their computational complexity increases exponentially with time. In this article, we propose a variational approximation of piecewise linear dynamical systems. We provide full details of the derivation of two variational expectation-maximization algorithms: a filter and a smoother. We show that the model parameters can be split into two sets: static and dynamic parameters, and that the former parameters can be estimated offline together with the number of linear modes, or the number of states of the switching variable. We apply the proposed method to the head-pose tracking, and we thoroughly compare our algorithms with several state of the art trackers.

Pseudo-Bautin Bifurcation in 3D Filippov Systems

Consider a piecewise linear dynamical system in [Formula: see text], divided by a switching plane, with the Teixeira Singularity (TS). Under the antiparallelism hypothesis, it is known that the system undergoes the so-called pseudo-Hopf bifurcation (pH). In this paper, we consider a particular position of the antiparallel vectors to yield a two-parametric unfolding of the TS. The bifurcation analysis gives us, besides the curve of pH bifurcation points, a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin bifurcation, since it exhibits the exact same behavior as the Bautin bifurcation for smooth dynamical systems.

Three Crossing Limit Cycles in a 3D-Filippov System Having a T-Singularity

Our starting point is a discontinuous piecewise linear dynamical system in [Formula: see text] with two zones that present a single invisible two-fold straight line and invariant cylinders. We broke the dynamics of this vector field to obtain a new four-parameter piecewise linear vector field with a [Formula: see text]-singularity. For this vector field, we prove that the upper bound of simple crossing limit cycles (simple CLCs) is three, and provide conditions for the existence of 1, 2, or 3 simple CLCs. We also show the occurrence of three distinct bifurcations involving such simple CLCs that are derived from a bifurcation set with two parameters: (i) a Teixeira Singularity bifurcation (TS-bifurcation); (ii) a Fold bifurcation; and (iii) a Cusp bifurcation.

Non-Uniqueness of Cycles in Piecewise-Linear Models of Circular Gene Networks

We find conditions for existence of two cycles for a five-dimensional piecewise-linear dynamical system that models functioning of a circular gene network. Conditions for existence of a cycle were obtained by the authors earlier. The phase portrait of a system is divided into subdomains (or blocks). With the use of such a discretization, we construct a combinatorial scheme for passages of trajectories between blocks. For the second cycle, we show that such a scheme depends on the parameters of a system.

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.

Structure of the phase portrait of a piecewise-linear dynamical system

Structure of the phase portrait of a piecewise-linear dynamical system

Poincaré Maps of “ <”-Shape Planar Piecewise Linear Dynamical Systems with a Saddle

It is important in the study of limit cycles to investigate the properties of Poincaré maps of discontinuous dynamical systems. In this paper, we focus on a class of planar piecewise linear dynamical systems with “[Formula: see text]”-shape regions and prove that the Poincaré map of a subsystem with a saddle has at most one inflection point which can be reached. Furthermore, we show that one class of such systems with a saddle-center has at least three limit cycles; a class of such systems with saddle and center in the normal form has at most one limit cycle which can be reached; and a class of such systems with saddle and center at the origin has at most three limit cycles with a lower bound of two. We try to reveal the reasons for the increase of the number of limit cycles when the discontinuity happens to a system.

Structure of the Phase Portrait of a Piecewise-Linear Dynamical System

Structure of the Phase Portrait of a Piecewise-Linear Dynamical System

A Class of Piecewise Linear Systems Without Equilibria With 3-D Grid Multiscroll Chaotic Attractors

In this brief a new class of piecewise linear dynamical system without equilibria which exhibits a 3-D grid multiscroll chaotic attractor is presented. The number of scrolls of the attractor generated can be easily changed by the number of linear parts of three piecewise constant functions. A particular system with a 3-D grid multiscroll attractor whose scrolls appear in an arrangement of ${3}{\times }{3}{\times }{3}$ is taken as a case study. Moreover, an electronic circuit realization is proposed for the particular system and simulation data as well as experimental data is provided.

Monotonicity of the Poincaré Mapping in Some Models of Circular Gene Networks

We obtain some sufficient conditions for the existence of a periodic trajectory of the Elowitz-Leibler type piecewise linear dynamical system that simulates a simplest nonsymmetric circular gene network. We prove the monotonicity of the corresponding Poincare mapping and construct an invariant toric neighborhood of this cycle.

Identification of piecewise linear dynamical systems using physically-interpretable neural-fuzzy networks: Methods and applications to origami structures

Self-locking origami structures are characterized by their piecewise linear constitutive relations between force and deformation, which, in practice, are always completely opaque and unmeasurable: the number of piecewise segments, the positions of non-smooth points, and the linear parameters of each segment are unknown a priori. However, acquiring this information is of fundamental importance for understanding the origami structure’s dynamic folding process and predicting its dynamic behaviors. This, therefore, arouses our interest in adopting a dynamical identification process to determine the model and to estimate the parameters. In this research, based on the piecewise linear assumption, a physically-interpretable neural-fuzzy network is built to correlate the measured input and output data. Unlike the conventional approaches, the constructed neural network possesses specific physical meaning of its components: the number of neurons relates to the number of piecewise segments, the coefficients of the local linear models relate to the parameters of the constitutive relations, and the validity functions relate to the positions of non-smooth points. By addressing several examples with different backgrounds, the network’s underlying data training methods are illustrated, including the local linear optimization for linear parameters, nested optimization for nonlinear partitions, and Local Linear Model Tree optimization for model selection. Noting that the tackled origami problem holds strong universality in terms of the unknown piecewise characteristics, the proposed approach would thus provide an effective, generic, and physically significant means for handling piecewise linear dynamical systems and meanwhile bring fresh vitality to the artificial neural network research.

Limit Cycles of a Class of Discontinuous Planar Piecewise Linear Systems with Three Regions of Y-Type

In this paper, we investigate the crossing limit cycles of a class of discontinuous planar piecewise linear dynamical systems with three regions of Y-type. We want to find the reason that cause the increase of the number of such limit cycles of the piecewise systems. We obtain the canonical form of such systems with focus–focus–focus type, and prove that the graph of Poincare map of one of three subsystems has one inflection point. Naturally, we prove that the lower bound of the number of crossing limit cycles for this class systems is 3. We also give an example to simulate the four real crossing limit cycles.

On the Teixeira singularity bifurcation in a DC–DC power electronic converter

The electronic circuit of a DC–DC boost power converter under a specific sliding mode control strategy is analyzed. This circuit is modeled as a discontinuous piecewise linear three-dimensional dynamical system, whose state space is divided into two open regions by a plane acting as the switching manifold. This system displays sliding motion confined to the switching manifold and limited by two straight lines where lie the points of tangency, one for each involved linear vector field. Such tangency lines can intersect transversally on a invisible two-fold point known as Teixeira singularity. Our goal is to investigate an interesting bifurcation that occurs on the Teixeira singularity, involving both a pseudo-equilibrium point and a crossing periodic orbit. This bifurcation, named TS-bifurcation, occurs when a pseudo-equilibrium point of the discontinuous piecewise linear system, capable of moving between the attractive and repulsive sliding regions as a result of the change in a parameter, collides with the Teixeira singularity. Simultaneously, a periodic orbit arises from the Teixeira singularity crossing the switching manifold in two points. Apart from the TS-bifurcation characterization in the DC–DC converter, we have numerically detected other non-local phenomena like a saddle-node bifurcation of crossing limit cycles. Experimental results to illustrate the effects of the TS-bifurcation on the dynamics of a power electronic converter are also shown.

An efficient method for simulating the dynamic behavior of periodic structures with piecewise linearity

An efficient method based on the parametric variational principle (PVP) is proposed for simulating the dynamic behavior of periodic structures with large number of piecewise linearity. The formulation for the gap-activated springs is proposed based on PVP, which can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method, an efficient numerical time-integration method is developed to obtain the dynamic responses of the system. For this method, the matrix exponential of only one unit cell of the system is computed, which greatly improves the computational efficiency. Dynamic responses of a 3 degrees of freedom (DOF) piecewise linear system under harmonic excitations are given to demonstrate the validation of the proposed method. The piecewise linear dynamic system can exhibit very complex vibrational behavior, such as stable periodic motion, multi-periodic motion, quasi-periodic motion and chaotic motion, which can be successfully predicted by using bifurcation theory. Moreover, it is demonstrated that the proposed method can be used to efficiently determine dynamic responses of a periodic piecewise linear system with large number of DOFs and large number of gap-activated springs under harmonic excitations.

A multiple focus-center-cycle bifurcation in 4D discontinuous piecewise linear memristor oscillators

The dynamical richness of 4D memristor oscillators has been recently studied in several works, showing different regimes, from stable oscillations to chaos. Typically, only numerical simulations have been reported and so there is a lack of mathematical results. We focus our analysis in the existence of multiple stable oscillations in the 4D piecewise linear version of the canonical circuit proposed by Itoh and Chua (Int J Bifurc Chaos 18(11):3183–3206, 2008). This oscillator is modeled by a discontinuous piecewise linear dynamical system. By adding one parameter that stratifies the 4D dynamics, it is shown that the dynamics in each stratum is topologically equivalent to a 3D continuous piecewise linear dynamical system. Some previous results on bifurcations in such reduced system allow to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits.

Simple agent-based dynamical system models for efficient financial markets: Theory and examples

We propose an agent-based framework, based on simple piecewise linear time-invariant continuous-time dynamical systems models, as a means for describing efficient financial markets. We show by examples that many of the common agent-specific trading strategies occurring in the academic literature, including chartists and fundamentalists of various kinds, can be described in the proposed framework. We present definitions for weak and strong market efficiency and provide necessary and sufficient conditions for them to hold. We present minimal examples of strongly and weakly efficient markets to show that these concepts are natural and easy to satisfy in agent-based models, and that the models can reproduce both statistical and behavioral stylized facts of real markets. We provide examples to demonstrate that the framework can be extended for agents with delays in information processing, as well as for agents with time-varying strategies and for nonlinear market impact functions. We also provide a counterexample to show that the proposed market efficiency concepts may require modification in generalizations for nonlinear trading strategies.

Dynamics of Discrete Time Systems with a Hysteresis Stop Operator

We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called a hysteresis stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. More complicated systems involving stop operators may have potential applications in the economic modeling of agents that display frictions and memory dependence.

A Piecewise-Linear Model of Intracranial Pressure Dynamics

Elevated intracranial pressure (ICP) is an extremely dangerous condition for patients suffering from traumatic brain injury, hydrocephalus, or related neurological disorders. To make informed decisions when treating such patients, clinicians must understand how the body controls ICP by regulating the rates at which cerebrospinal fluid (CSF) is formed and reabsorbed. Mathematical models can aid in this task. Of particular interest are models that can help us understand the transition between the stable, approximately constant values of ICP found in healthy individuals, and pathological oscillatory behaviors such as those observed with plateau waves, in which ICP exhibits steady oscillations between high and low pressures. In this paper, we develop a mathematical model of ICP dynamics with the goal of illustrating how the transition to oscillatory ICP dynamics can arise from processes that regulate CSF formation. A simple low-dimensional model is built that couples brain tissue mechanics and CSF hydraulics. Balance of mass and linear momentum result in a damped linear oscillator in terms of a single configuration variable representing deformation of brain tissue caused by changes in ICP. We focus on the case where the CSF supply rate is regulated by a piecewise mechanism based on the total volume of CSF. We show that the resulting piecewise-linear dynamical system can exhibit limit cycles in a manner consistent with plateau waves. We conclude by physically interpreting our results and discussing their potential clinical implications.

On the Convergence of Piecewise Linear Strategic Interaction Dynamics on Networks

We prove that the piecewise linear best-response dynamical systems of strategic interactions are asymptotically convergent to their set of equilibria on any weighted undirected graph. We study various features of these dynamical systems, including the uniqueness and abundance properties of the set of equilibria and the emergence of unstable equilibria. We also introduce the novel notions of social equivalence and social dominance on directed graphs, and demonstrate some of their interesting implications, including their correspondence to consensus and chromatic number of partite graphs. Examples illustrate our results.