Piecewise Smooth Dynamical Systems Research Articles

Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.

Non-Filippov dynamics arising from the smoothing of nonsmooth systems, and its robustness to noise

Switch-like behaviour in dynamical systems may be modelled by highly nonlinear functions, such as Hill functions or sigmoid functions, or alternatively by piecewise-smooth functions, such as step functions. Consistent modelling requires that piecewise-smooth and smooth dynamical systems have similar dynamics, but the conditions for such similarity are not well understood. Here we show that by smoothing out a piecewise-smooth system one may obtain dynamics that is inconsistent with the accepted wisdom — so-called Filippov dynamics — at a discontinuity, even in the piecewise-smooth limit. By subjecting the system to white noise, we show that these discrepancies can be understood in terms of potential wells that allow solutions to dwell at the discontinuity for long times. Moreover we show that spurious dynamics will revert to Filippov dynamics, with a small degree of stochasticity, when the noise magnitude is sufficiently large compared to the order of smoothing. We apply the results to a model of a dry-friction oscillator, where spurious dynamics (inconsistent with Filippov’s convention or with Coulomb’s model of friction) can account for different coefficients of static and kinetic friction, but under sufficient noise the system reverts to dynamics consistent with Filippov’s convention (and with Coulomb-like friction).

The Role of Constraints in a Segregation Model: The Asymmetric Case

We extend the analysis on the effects of the entry constraints on the dynamics of an adaptive segregation model of Shelling’s type when the two populations involved differ in numerosity, level of tolerance toward members of the other population, and speed of reaction. The model is described by a two-dimensional piecewise smooth dynamical system in discrete time, where the entry constraints represent possible exogenous controls imposed by an authority in order to regulate the maximum number of members of the two populations allowed to enter the system, usually the district in which they live in. In this paper, we investigate the nature of some particular border collision bifurcations and discuss the policy implications of the entry constraints in terms of segregation. The investigation reveals that asymmetries in the level of tolerance of the two populations involved may lead to phenomena of overreaction or overshooting in the adjustment process. In order to avoid the risk of segregation, suitable entry limitations must be imposed at least on the more tolerant population.

Contraction Analysis for a Class of NonDifferentiable Systems with Applications to Stability and Network Synchronization

In this paper we extend to a generic class of piecewise smooth dynamical systems a fundamental tool for the analysis of convergence of smooth dynamical systems: contraction theory. We focus on switched nondifferentiable systems satisfying Carathéodory conditions for the existence and uniqueness of a solution. After generalizing the classical definition of contraction to this class of dynamical systems, we give sufficient conditions for global convergence of their trajectories. The theoretical results are then applied to solve a set of representative problems such as proving global asymptotic stability of switched linear systems, giving conditions for incremental stability of piecewise smooth systems, and analyzing the convergence of networked switched systems.

Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools

The isolated spectrum of transfer operators is known to play a critical role in determining mixing properties of piecewise smooth dynamical systems. The so-called Dellnitz-Froyland ansatz places isolated eigenvalues in correspondence with structures in phase space that decay at rates slower than local expansion can account for. Numerical approximations of transfer operator spectrum are often insufficient to distinguish isolated spectral points, so it is an open problem to decide to which eigenvectors the ansatz applies. We propose a new numerical technique to identify the isolated spectrum and large-scale structures alluded to in the ansatz. This harmonic analytic approach relies on new stability properties of the Ulam scheme for both transfer and Koopman operators, which are also established here. We demonstrate the ecacy of this scheme in metastable one- and two-dimensional dynamical systems, including those with both expanding and contracting dynamics, and explain how the leading eigenfunctions govern the dynamics for both real and complex isolated eigenvalues.

The Border Collision Normal Form with Stochastic Switching Surface

In the study of discrete time piecewise smooth dynamical systems the deterministic border collision normal form describes the bifurcations as a fixed point moves across the switching surface with changing parameter. If the position of the switching surface varies randomly, we give conditions which imply that the attractor close to the bifurcation point is the attractor of an iterated function system. The proof uses an equivalent metric to the Euclidean metric because the functions involved are never contractions in the Euclidean metric. If the conditions do not hold, then a range of possibilities may be realized, including local instability, and some examples are investigated numerically.

High-Feedback Operation of Power Electronic Converters

The purpose of this review is to provide a survey of some of the most important bifurcation phenomena that one can observe in pulse-modulated converter systems when operating with high corrector gain factors. Like other systems with switching control, electronic converter systems belong to the class of piecewise-smooth dynamical systems. A characteristic feature of such systems is that the trajectory is “sewed” together from subsequent discrete parts. Moreover, the transitions between different modes of operation in response to a parameter variation are often qualitatively different from the bifurcations we know for smooth systems. The review starts with an introduction to the concept of border-collision bifurcations and also demonstrates the approach by which the full dynamics of the piecewise-linear, time-continuous system can be reduced to the dynamics of a piecewise-smooth map. We describe the main bifurcation structures that one observes in three different types of converter systems: (1) a DC/DC converter; (2) a multi-level DC/DC converter; and (3) a DC/AC converter. Our focus will be on the bifurcations by which the regular switching dynamics becomes unstable and is replaced by ergodic or resonant periodic dynamics on the surface of a two-dimensional torus. This transition occurs when the feedback gain is increased beyond a certain threshold, for instance in Electronics 2013, 2 114 order to improve the speed and accuracy of the output voltage regulation. For each of the three converter types, we discuss a number of additional bifurcation phenomena, including the formation and reconstruction of multi-layered tori and the appearance of phase-synchronized quasiperiodicity. Our numerical simulations are compared with experimentally observed waveforms.

Stability conditions in piecewise smooth dynamical systems at a two-fold singularity

Some qualitative and geometric aspects of 3-dimensional non-smooth vector fields theory are discussed. Our main aim is to study the dynamics near typical singularities of piecewise smooth dynamical systems, the so-called two-fold singularities. More specifically, we are interested in discussing stability problems of such systems around these singularities.

Complex bifurcations in a nonlinear system of moving belt

A kind of one-degree-of-freedom nonlinear moving belt system is considered. The analytical research of sliding region and existence conditions of equilibrium are first derived by the theory of piecewise-smooth dynamical system. Then, using numerical method, one- or two-parameter continuation of several types of periodic orbits of the system is calculated. We obtain codimension-1 sliding bifurcation curves, codimension-2 sliding bifurcation points, and global bifurcation diagram in parameter space for the system. The investigation of bifurcation behavior shows that the speed of moving belt and amplitude of friction have a great influence on dynamic behavior, and reveals the complex nonlinear dynamic phenomenon of the moving belt system.

Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems

We investigate a generalized Hopf bifurcation emerged from a corner located at the origin which is the intersection of n discontinuity boundaries in planar piecewise smooth dynamical systems with the Jacobian matrix of each smooth subsystem having either two different nonzero real eigenvalues or a pair of complex conjugate eigenvalues. We obtain a novel result that the generalized Hopf bifurcation can occur even when the Jacobian matrix of each smooth subsystem has two different nonzero real eigenvalues. According to the eigenvalues of the Jacobian matrices and the number of smooth subsystems, we provide a general method and prove some generalized Hopf bifurcation theorems by studying the associated Poincaré map.

Fluid limit of an asynchronous optical packet switch with shared per link full range wavelength conversion

We consider an asynchronous all optical packet switch (OPS) where each link consists of N wavelength channels and a pool of C ≤ N full range tunable wavelength converters. Under the assumption of Poisson arrivals with rate λ (per wavelength channel) and exponential packet lengths, we determine a simple closed-form expression for the limit of the loss probabilities P loss (N) as N tends to infinity (while the load and conversion ratio σ=C/N remains fixed). More specifically, for σ ≤ λ 2 the loss probability tends to (λ 2 -σ)/λ(1+λ), while for σ > λ 2 the loss tends to zero. We also prove an insensitivity result when the exponential packet lengths are replaced by certain classes of phase-type distributions. A key feature of the dynamical system (i.e., set of ODEs) that describes the limit behavior of this OPS switch, is that its right-hand side is discontinuous. To prove the convergence, we therefore had to generalize some existing result to the setting of piece-wise smooth dynamical systems.

Noise-induced Phenomena in Two Strongly Pulse-coupled Spiking Neurons

In this study, we address noise-induced phenomena in a system of two pulse-coupled resonate-and-fire neuron models as a piecewise smooth dynamical system. The model is a spiking neuron model that has second-order membrane dynamics with a firing threshold, and it exhibits subthreshold oscillation of membrane potential, resulting in the sensitivity to the timing of inputs. The pulse-coupled system shows the grazing bifurcation, a specific feature of a piecewise smooth dynamical system, as well as the saddle-node bifurcation. We numerically investigated mixture bifurcation structure of the system, leading to complex behaviors, such as burst synchronization and chaotic phenomena. Moreover, we found that a novel type of noise-induced order and chaos occur in the system when additive noise are given as a perturbation on the firing threshold. Finally, we will discuss a considerable mechanism of such noise-induced phenomena and the relationship to discontinuity-induced bifurcation in view of the piecewise smooth dynamical system theory.

Modeling and Numerical Simulation of the Nonlinear Dynamics of the Parametrically Forced String Pendulum

The string pendulum consists of a mass attached to the end of an inextensible string which is fastened to a support. Applying an external forcing to the pendulum’s support is motivated by understanding the behavior of suspension bridges or of tethered structures during earthquakes. The forced string pendulum can go from taut to slack states and vice versa, and is capable of exhibiting interesting periodic and chaotic dynamics. The inextensibility of the string and its capacity to go slack make simulation and analysis of the system complicated. The string pendulum system is thus formulated here as a piecewise-smooth dynamical system using the method of Lagrange multipliers to obtain a system of differential algebraic equations (DAE) for the taut state. In order to develop a formulation for the forced string pendulum system, we first turn to similar but simpler pendulum systems, such as the classic rigid pendulum, the elastic spring pendulum and the elastic spring pendulum with piecewise constant stiffness. We perform a perturbation analysis for both the unforced and forced cases of the spring pendulum approximation, which shows that, for large stiffness, this is a reasonable model of the system. We also show that the spring pendulum with piecewise constant stiffness can be a good approximation of the string pendulum, in the limit of a large extension constant and a low compression constant. We indicate the behavior and stability of this simplified model by using numerical computations of the system’s Lyapunov exponents. We then provide a comparison of the spring pendulum with piecewise constant stiffness with the formulation of the taut-slack pendulum using the DAE for the taut states and derived switching conditions to the slack state.

On 3-Parameter Families of Piecewise Smooth Vector Fields in the Plane

This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of non$-$smooth vector fields are studied and the tridimensional bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called fold-cusp singularity by means of the variation of 3 parameters.

On the number of limit cycles in general planar piecewise linear systems

Much progress has been made in planar piecewise smooth dynamical systems. However there remain many important problems to be solved even in planar piecewise linear systems. In this paper, we investigate the number of limit cycles of planar piecewise linear systems with two linear regions sharing the same equilibrium. By studying the implicit Poincaré map induced by the discontinuity boundary, some cases when there exist at most 2 limit cycles is completely investigated. Especially, based on these results we provide an example along with numerical simulations to illustrate the existence of 3 limit cycles thus have a negative answer to the conjecture by M. Han and W. Zhang [11](J. Differ.Equations 248 (2010) 2399-2416) that piecewise linear systems with only two regions have at most 2 limit cycles.

Continuation of oscillatory patterns in a road traffic model of three cars

AbstractWe investigate microscopic models of the road traffic. In particular, we consider the car-following model for a single-line traffic flow of N identical cars on a circular road. The classical differentiable model breaks down at the time instant when two cars collide. Nevertheless, the natural action of a driver would be to overtake the slower car. In our previous work, we proposed the model which simulates overtaking. We observed a large variety of oscillatory solutions (oscillatory patterns) of the model. In the present contribution, we assume N=3 i.e., three cars on the route, and formulate our model as a particular piecewise smooth (PWS) dynamical system. We define generic oscillatory patterns as invariant objects of the flow defined by this PWS system. We use the standard software to continue the patterns with respect to a parameter (the length of the route). In this respect, we also investigate bifurcation phenomena.

Slow–fast systems on algebraic varieties bordering piecewise-smooth dynamical systems

This article extends results contained in Buzzi et al. (2006) [4] , Llibre et al. (2007, 2008) [12] , [13] concerning the dynamics of non-smooth systems. In those papers a piecewise C k discontinuous vector field Z on R n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F : U → R a polynomial function defined on the open subset U ⊂ R n . The set F − 1 ( 0 ) divides U into subdomains U 1 , U 2 , … , U k , with border F − 1 ( 0 ) . These subdomains provide a Whitney stratification on U . We consider Z i : U i → R n smooth vector fields and we get Z = ( Z 1 , … , Z k ) a discontinuous vector field with discontinuities in F − 1 ( 0 ) . Our approach combines several techniques such as ε -regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an ε -regularization of Z (see Sotomayor and Teixeira, 1996 [18] ; Llibre and Teixeira, 1997 [15] ). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16] ), in systems with hysteresis (Seidman, 2006 [17] ) and in mechanical systems with impacts (di Bernardo et al., 2008 [5] ).

Hamiltonian flows with random-walk behaviour originating from zero-sum games and fictitious play**To Jacob Palis on his 70th birthday.

In this paper we introduce Hamiltonian dynamics, inspired by zero-sum games (best response and fictitious play dynamics). The Hamiltonian functions we consider are continuous and piecewise affine (and of a very simple form). It follows that the corresponding Hamiltonian vector fields are discontinuous and multi-valued. Differential equations with discontinuities along a hyperplane are often called ‘Filippov systems’, and there is a large literature on such systems, see for example (di Bernardo et al 2008 Theory and applications Piecewise-Smooth Dynamical Systems (Applied Mathematical Sciences vol 163) (London: Springer); Kunze 2000 Non-Smooth Dynamical Systems (Lecture Notes in Mathematics vol 1744) (Berlin: Springer); Leine and Nijmeijer 2004 Dynamics and Bifurcations of Non-smooth Mechanical Systems (Lecture Notes in Applied and Computational Mechanics vol 18) (Berlin: Springer)). The special feature of the systems we consider here is that they have discontinuities along a large number of intersecting hyperplanes. Nevertheless, somewhat surprisingly, the flow corresponding to such a vector field exists, is unique and continuous. We believe that these vector fields deserve attention, because it turns out that the resulting dynamics are rather different from those found in more classically defined Hamiltonian dynamics. The vector field is extremely simple: outside codimension-one hyperplanes it is piecewise constant and so the flow ϕt piecewise a translation (without stationary points). Even so, the dynamics can be rather rich and complicated as a detailed study of specific examples show (see for example theorems 7.1 and 7.2 and also (Ostrovski and van Strien 2011 Regular Chaotic Dynf. 16 129–54)). In the last two sections of the paper we give some applications to game theory, and finish with posing a version of the Palis conjecture in the context of the class of non-smooth systems studied in this paper.

Canards and curvature: nonsmooth approximation by pinching

In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blow-up. More recently, canard-like behaviour has been linked to surfaces of discontinuity in piecewise-smooth dynamical systems.This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewise-smooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all time-scales. The changes of local curvature that are responsible for canard explosions are then analysed. Regions where different time-scales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewise-smooth system, in which curvature changes manifest as discontinuity-induced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head).

Existence and stability of limit cycles in a delayed dry-friction oscillator

Periodic, chaotic, chattering, and bifurcation behavior are fundamental consequences of the nonsmooth nature of systems with dry friction. This work is concerned with the analysis of a single degree of freedom system which is additionally damped by a delayed dry friction device. We get a complete set of closed-form expressions to describe the dynamics of the delay-induced phenomena exhibited by the system. The conditions to determine the existence and stability of limit cycles are clearly defined. This analysis is addressed in the context of both classic stability theory for nonlinear systems and the qualitative theory of Piecewise Smooth Dynamical Systems. Through exhaustive numerical simulations the effectiveness of the set of closed-form expressions is confirmed. Excellent agreement was found between the numerical and analytical results.