Piecewise Smooth Dynamical Systems Research Articles

Bifurcation analysis of piecewise-smooth engineering systems with delays through numeric continuation of periodic orbits

This paper presents a numeric continuation framework for periodic orbits of piecewise-smooth and hybrid dynamical systems with fixed point delays. For the numeric solution of the corresponding infinite dimensional multi-point boundary value problem, a novel discretization and interpolation scheme is developed employing Chebyshev polynomial based spectral collocation techniques. The same approach is employed for the formulation of the corresponding monodromy matrix enabling stability analysis on the found periodic orbits. Special care is attributed to the accurate detection of discontinuity induced bifurcations such as grazing and sliding, and the implemented pseudo-arclength framework is adapted to allow two parameter continuation of these critical points. The capabilities of the developed algorithms are demonstrated on a set of delayed piecewise-smooth and hybrid dynamical systems, showcasing potential engineering applications from the fields of control theory, traffic dynamics modelling, and machine tool vibrations. Finally, a detailed tutorial is attached in the appendix to accompany the open-source release of the developed codebase.

Fold-Fold Singularity in a Piecewise Smooth Mathematical Model Describing the Dynamics of a Stockless Market

Fold-fold singularities are critical points or singularities in piecewise smooth dynamical systems (PWS) where both the stability and the structure of the system change. These singularities are of great importance in the study of specific dynamics, such as those in markets, as they indicate significant transformations in their evolution, including sudden variability in prices or changes in the behavior of offers and demand. Despite the substantial increase in the use of mathematical and computational tools applied to market dynamics, the current literature does not thoroughly address the study of the existence of fold-fold singularities in piecewise smooth systems within this context. Therefore, due to the importance of markets as economic activities, this paper proves the existence of such a singularity in a mathematical model that describes the dynamics of a stockless market, which is represented by a system of ordinary differential equations defined with piecewise smooth functions.

Periodic orbits for 2n-dimensional control piecewise smooth dynamical systems

Periodic orbits for 2n-dimensional control piecewise smooth dynamical systems

Model reduction to spectral submanifolds in piecewise smooth dynamical systems

We develop a model reduction technique for piecewise smooth dynamical systems using spectral submanifolds. Specifically, we construct low-dimensional, sparse, nonlinear and piecewise smooth models on unions of slow and attracting spectral submanifolds (SSMs) for each smooth subregion of the phase space and then properly match them. We applied this methodology to both equation-driven and data-driven problems, with and without external forcing.

On the Computation of Floquet Multipliers for Periodic Solution in Piecewise-smooth Dynamical System

The floquet multiplier is one of the most important indicators for the stability and bifurcation analysis for periodic solutions in nonlinear dynamical systems. Different from the well-established Floquet theory for the perturbation systems of smooth systems, much less has been understood in its counterpart for non-smooth systems. Here in this paper, we will report an unusual and interesting feature of the Floquet multipliers for piecewise-smooth dynamical systems. When the initial condition of the periodic solution is located at the boundary splitting the solution domain, the multipliers would be calculated falsely in certain circumstances, respectively, by a saltation matrix method or a direct numerical integration for the perturbation system. We elucidate the origin of the fake multipliers through perturbation analysis, and furthermore suggest an effective manner to avoid the miscalculation. This finding would be of fundamental significance to both the real-world applications and theory establishment of the Floquet theory in non-smooth systems

Fast-slow dynamics analysis in an externally excited smooth and discontinuous oscillator with a pair of irrational nonlinearities

The study of fast-slow oscillations in systems with irrational nonlinearity that may yield abundant dynamical mechanisms is not well developed. This paper aims to investigate the fast-slow dynamics in an excited mass-spring oscillator with a pair of irrational nonlinearities, which can undergo the dynamical transition from smooth to discontinuous characteristics depending on the values of a smoothness parameter. Three different types of fast-slow oscillations are reported in this interesting smooth and discontinuous (SD) oscillator with a pair of irrational nonlinearities. Due to the smooth and discontinuous characteristics of this SD oscillator, we consider its dynamical behaviors under the smooth and discontinuous cases, respectively. Based on the fast-slow analysis and the two-parameter bifurcation analysis, the smooth fast-slow dynamics associated with fold hysteresis and its turnover are revealed. In the discontinuous case, the system can be viewed as a piecewise-smooth dynamical system governed by three different subsystems in different regions divided by two nonsmooth boundaries. In particular, the nonsmooth boundaries can be divided into parts with different dynamical behaviors, including escaping and crossing lines. Unlike the smooth case, there is no change in the stability of the equilibrium in these three subsystems. However, transitions of system trajectory induced by crossing lines can account for the generation of fast-slow oscillations in the piecewise-smooth system. As a result, the smooth and piecewise-smooth fast-slow dynamics in the excited SD oscillator with a pair of irrational nonlinearities are revealed, which deepens the understanding of fast-slow dynamics of the dynamical systems with irrational nonlinearity.

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.

Effects of PLL Frequency Limiters on Synchronization Stability of Grid Connected VSC and Strategy to Realize Global Stability

This paper investigates the synchronization stability of voltage source converter (VSC) connected to infinite bus system with frequency limiter of phase-locked loop (PLL). First, three different types of PLL frequency limiters are modelled based on piecewise-smooth dynamical system (PSDS). Then the effects of different frequency limiters and frequency limit values on the region of attraction (ROA) of the system are studied and the properties of the PSDSs are analyzed. It is found that PLL with anti-windup limiter with back calculation can achieve global stability with appropriate limit value, and the principle of setting limit value based on a specific ROA estimation is proposed. Based on ROA estimation with Lyapunov function, a method of setting frequency limit value which guarantees global stability is proposed. The validity of the proposed strategy is verified through MATLAB/Simulink simulation.

Vibro-impact dynamics of an experimental rig with two-sided constraint and bidirectional drift

This paper demonstrates a theoretical and experimental study on a universal vibro-impact system with bidirectional drift. With a dedicated experimental rig that is able to verify numerical simulations, the system has revealed various bifurcations from a piecewise-smooth dynamical model involving impact and dry friction. This experimental rig with a two-sided constraint is designed with adjustable configurations which can be used for experimental testing of versatile piecewise-smooth dynamical systems, e.g., the impact oscillator, the drifting oscillator and the capsule system. It is able to be customised to alter different system parameters, such as friction, number of constraints, mass ratio, impact gap and constraint stiffness, for developing and testing various control strategies. In this paper we particularly studied several types of bifurcations from our preliminary experimental results and discussed the dynamical responses of the system under different excitation frequencies and amplitudes. It was found that with a higher excitation frequency and amplitude, the vibro-impact dynamics on both ends of the constraints could result in a large amount of uncertainties, thus bi-stable and chaotic motions are likely to be observed. While being unstable, the system runs with a relatively high energy efficiency in this configuration.

Dynamics in sliding set of planar sector-wise linear systems

For piecewise smooth dynamical systems the existence and properties of some special kinds of sliding points are very important to determine the dynamics (whether local or global), and are also very essential in studying DIBs (i.e. discontinuity induced bifurcations) related to sliding motions. In this paper, we mainly study the inner dynamics of the sliding set of a general planar sector-wise linear differential systems, including the definitions, existence and stability of all special sliding points. Specially, by studying the case when the two zones are separated by straight lines, we obtain explicit dependence on system parameters of the existence, stability and number of all kinds of special sliding points for the planar sector-wise linear systems. Moreover, we provide concrete examples to illustrate our main results and their application in studying DIBs.

Uncountably Many Cases of Filippov’s Sewed Focus

The sewed focus is one of the singularities of planar piecewise smooth dynamical systems. Defined by Filippov in his book (Differential Equations with Discontinuous Righthand Sides, Kluwer, 1988), it consists of two invisible tangencies either side of the switching manifold. In the case of analytic focus-like behaviour, Filippov showed that the approach to the singularity is in infinite time. For the case of non-analytic focus-like behaviour, we show that the approach to the singularity can be in finite time. For the non-analytic sewed centre-focus, we show that there are uncountably many different topological types of local dynamics, including cases with infinitely many stable periodic orbits, and show how to create systems with periodic orbits intersecting any bounded symmetric closed set.

Sliding modes of high codimension in piecewise-smooth dynamical systems

We consider piecewise-smooth dynamical systems, i.e., systems of ordinary differential equations switching between different sets of equations on distinct domains, separated by hyper-surfaces. As is well-known, when the solution approaches a discontinuity manifold, a classical solution may cease to exist. For this reason, starting with the pioneering work of Filippov, a concept of weak solution (also known as sliding mode) has been introduced and studied. Nowadays, the solution of piecewise-smooth dynamical systems in and close to discontinuity manifolds is well understood, if the manifold consists locally of a single discontinuity hyper-surface or of the intersection of two discontinuity hyper-surfaces. The present work presents partial results on the solution in and close to discontinuity manifolds of codimension 3 and higher.

Sliding fast–slow dynamics in the slowly forced Duffing system with frequency switching

This paper aims to report sliding fast–slow dynamics related to the classic relaxation oscillations. To this end, the slowly forced Duffing system with frequency switching is taken as an example. Typical fast–slow oscillations, i.e., the classic relaxation oscillations related to an S-shaped equilibrium hysteresis curve with two fold points, can be observed in the slowly forced Duffing system. We show that, in the presence of frequency switching, the fast–slow oscillations may exhibit distinct sliding behaviors, characterized by that the oscillation trajectory slides along the frequency-switching threshold for some time. Then, the frequency-switching threshold is taken as the control parameter. As a result, six different patterns of sliding fast–slow oscillations are obtained, and their dynamical mechanisms are revealed by theories of fast–slow dynamical systems and of piecewise-smooth dynamical systems. Our study shows that the frequency-switching threshold has great effects on dynamical characteristics of the switching boundary. In particular, the switching boundary can be divided into several areas whose dynamical characteristics may vary along with the frequency-switching threshold. Besides, dynamical behaviors of subsystems near the switching boundary depend heavily on the frequency-switching threshold. These factors account for the generation of different sliding fast–slow oscillation patterns in relation to the variation of frequency-switching threshold.

Chaos in One-dimensional Piecewise Smooth Dynamical Systems

Chaos in One-dimensional Piecewise Smooth Dynamical Systems

Chaotic Dynamics Arising from Sliding Heteroclinic Cycles in 3D Filippov Systems

Filippov systems are a representative class of piecewise smooth dynamical systems with sliding motion. It is known that such systems can exhibit complex dynamics, but how they generate chaos remains to be further studied. This paper establishes three Shilnikov-type heteroclinic theorems for 3-dimensional (3D) Filippov systems divided by a smooth surface, which admit heteroclinic cycles sliding on the switching surface. These theorems correspond to two typical scenarios of sliding heteroclinic cycles: (i) connecting two saddle-foci; (ii) connecting one saddle and one saddle-focus. In the presence of a sliding heteroclinic cycle, the corresponding Filippov system can be analytically proved to have a chaotic invariant set nearby the singular cycle under some assumed conditions. These results provide a reasonable explanation for the chaotic behaviors of 3D Filippov systems. Two numerical examples are presented to validate the theorems.

Bifurcation analysis of 3D-PWS systems with two transversal switching boundaries: A case study in power electronics

This paper addresses the nonlinear analysis of PWS dynamical systems with two transverse switching boundaries through a real case study: a cascade of two buck converters connected to a common bus in a dc microgrid. Two-parameter bifurcation analysis is proposed to study local and global dynamics on a power electronic circuit feeding a constant power load and controlled by means of a sliding mode control. This electronic circuit is modeled as a 3D piecewise smooth dynamical system with two transversal switching boundaries, having sliding motion in just one of them. Complete characterization of phenomena associated with the bifurcations induced by two transversal switching boundaries is one of the main contributions of this work. Numerical continuation methods are employed to obtain bifurcation sets and bifurcation diagrams to understand the dynamic behavior of the study system and also to verify the analytical results.

Sliding modes in the management of renewable resources

The presence of tipping points in ecological systems implies abrupt changes in the dynamics of the ecosystem. In these piecewise-smooth dynamical systems sliding dynamics, i.e., dynamics on the switching boundary, have been reported for population models. However, the question whether or not, and if so under which conditions, sliding dynamics may occur in an optimally controlled system have not yet been studied. We explore this issue in a simple harvesting model with two regimes, and find that optimal sliding may occur if regular steady states do not exist. Hence, sliding dynamics may be part of an optimal policy.

Steady-state dynamics and discontinuity-induced sliding bifurcation of a multi-module piecewise-smooth vibration-driven system with dry friction

The vibration-driven locomotion system subject to the Coulomb dry friction is a typical multi-degree-of-freedom piecewise-smooth dynamical system with rigid-body displacement. Rather than ignoring the sticking behaviors caused by dry friction in previous studies, this paper focuses on the steady-state dynamics and the discontinuity-induced sliding bifurcations of a non-smooth three-module vibration-driven system. It is shown that the presence of sticking partially invalidates the averaging method and prevents an accurate prediction of the locomotion performance. As a result, numerical methods are employed in this research, which not only gives the distribution pattern of the average steady-state velocity with respect to the actuation phase differences but also elucidates its variation trend with respect to the actuation frequency and friction anisotropism, revealing the significance of the resonance effect and phase coordination for locomotion performance enhancement. Subsequently, this paper focuses on the stick–slip dynamics associated with the non-smooth boundaries induced by dry friction. Through comprehensive numerical calculations, we identify four different types of stick–slip motions and their distributed zones, whose boundaries are essentially the sliding bifurcation curves. Analytical expressions of these bifurcation curves are derived by means of the least-square regression, and they together constitute a bifurcation diagram for a given actuation and environment conditions. Each time a sliding bifurcation curve is traversed, the stick–slip trajectories experience qualitative changes. We also systematically reveal the evolution of the bifurcation diagram with the actuation frequency and friction anisotropism.

Two regularizations of the grazing-sliding bifurcation giving non equivalent dynamics

We present two ways of regularizing a one parameter family of piece-wise smooth dynamical systems undergoing a codimension one grazing-sliding global bifurcation of periodic orbits. First we use the Sotomayor-Teixeira regularization and prove that the regularized family has a saddle-node bifurcation of periodic orbits. Then we perform a hysteretic regularization and show that the regularized family has chaotic dynamics. Our result shows that, in spite that the two regularizations will give the same dynamics in the sliding modes, when a tangency appears the hysteretic process generates chaotic dynamics.

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise smooth resistor and a harmonically changed electric source into a typical Chua’s circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation.