Piecewise Smooth Systems Research Articles

Stability and Periodic Motions for a System Coupled with an Encapsulated Nonsmooth Dynamic Vibration Absorber

The dynamic vibration absorber (DVA) is widely used in engineering models with complex vibration modes. The research on the stability and periodic motions of the DVA model plays an important role in revealing its complex vibration modes and energy transfer. The aim of this paper is to study the stability and periodic motions of a two-degrees-of-freedom system coupled with an encapsulated nonsmooth dynamic vibration absorber under low-frequency forced excitation. Based on the slow–fast method, the model is transformed into a six-dimensional piecewise smooth system coupling two time scales. The existence and stability of the admissible equilibrium points for the model are discussed under different parameter conditions. Based on the first integrals, the Melnikov vector function of the nonsmooth dynamic vibration absorber model is calculated. The existence and number of periodic orbits bifurcated from a family of periodic orbits under different parameters are discussed. The phase diagram configuration of periodic orbits is given based on numerical simulation. The results obtained in this paper offer a new perspective for vibration analysis and parameter control for nonsmooth dynamic vibration absorbers.

Analysis of point-contact models of the bounce of a hard spinning ball on a compliant frictional surface

Abstract Inspired by the turf–ball interaction in golf, this paper seeks to understand the bounce of a ball that can be modelled as a rigid sphere and the surface as supplying a viscoelastic contact force in addition to Coulomb friction. A general formulation is proposed that models the finite time interval of bounce from touch-down to lift-off. Key to the analysis is understanding transitions between slip and roll during the bounce. Starting from the rigid-body limit with an energetic or Poisson coefficient of restitution, it is shown that slip reversal during the contact phase cannot be captured in this case, which generalizes to the case of pure normal compliance. Yet, the introduction of linear tangential stiffness and damping does enable slip reversal. This result is extended to general weakly nonlinear normal and tangential compliance. An analysis using the Filippov theory of piecewise-smooth systems leads to an argument in a natural limit that lift-off while rolling is non-generic and that almost all trajectories that lift off do so under slip conditions. Moreover, there is a codimension-one surface in the space of incoming velocity and spin which divides balls that lift off with backspin from those that lift off with topspin. The results are compared with recent experimental measurements on golf ball bounce and the theory is shown to capture the main features of the data.

Finite time event-triggered consensus of variable-order fractional multi-agent systems

This paper focuses on the finite time consensus issue for variable-order fractional multi-agent systems (FMASs), where each agent is characterized by the piecewise-smooth systems. Firstly, the new variable-order fractional difference inequality and finite time stability theorem are developed, which greatly extend some existing results. Secondly, a distributed control protocol is presented, where a new triggering function and an internal dynamic variable are constructed for designing the dynamic event-triggered mechanism (DETM) to enlarge average event execution times and reduce the consumption of system resources. Thirdly, based on the developed variable-order fractional inequalities and finite time stability theorem, the considered systems achieve consensus within a finite time. Furthermore, it is proved that the Zeno phenomenon do not occur. Finally, the circuit realization approach for the designed control protocol is proposed and simulation results are provided to verify the correctness of theoretical analysis.

Melnikov-type method for chaos in a class of hybrid piecewise-smooth systems with impact and noise excitation under unilateral rigid constraint

The Melnikov method for chaos is developed in a class of hybrid piecewise-smooth systems with impact and noise excitation under unilateral rigid constraint, the corresponding unperturbed Hamiltonian system has the three-piecewise homoclinic orbit. This homoclinic orbit crosses the first and second switching manifolds continuously, then jumps symmetrically with x-axis on the third switching manifold through the rigid impact. It will not cross the third switching manifold and enter the next zone under unilateral rigid constraint. Especially, the perturbed trajectory crosses the first switching manifold through the elastic impact rule expressed by the reset map. Afterwards, the random non-smooth Melnikov function is obtained, the chaotic criteria with and without noise excitation are derived. Furthermore, the theoretical results obtained are applied to analyse the chaos in an actual non-smooth mechanical system with impacts, dry friction and noise excitation.

The discontinuous matching of two globally asymptotically stable crossing piecewise smooth systems in the plane do not produce in general a piecewise differential system globally asymptotically stable

The discontinuous matching of two globally asymptotically stable crossing piecewise smooth systems in the plane do not produce in general a piecewise differential system globally asymptotically stable

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.

Unveiling the Effects of Viscous Friction on the Full Annular Rubs in a Piecewise Smooth Rotor/Stator Rubbing System

Experimental evidences show that friction during rotor/stator rubbing may not always obey the Coulomb friction law, which is predominantly adopted in the studies of these kinds. This work is devoted to unveiling the effects of viscous damping in the friction model on forward and backward full annular rubbing responses in a piecewise smooth rotor/stator rubbing system. The effects induced by the viscous friction, which are exclusively different from those with pure Coulomb friction, may possibly be used as the signatures for identifying the presence/extent of viscous friction in rotor/stator rubbing tests. In this work, the boundaries of both forward and backward full annular rub motions are analytically derived based on the stability and bifurcation theory. Due to the introduction of viscous friction, the rotor/stator system changes from a piecewise smooth continuous system to a Fillipov one. The results show that the viscous friction has little influence on the region of backward full annular rubbing responses but has significant effects on the region of forward full annular rub motion. With the increase of the viscous friction, the parameter region of the forward full annular rub shrinks significantly, and the boundary of Hopf bifurcation changes increasingly from supercritical to subcritical. This leads to more complex behaviors in the global response characteristics in comparison with those of the rotor/stator rubbing system with Coulomb friction, for instance, two new quasiperiodic rubbing responses appear beside the quasiperiodic partial rub bifurcating from the synchronous full annular rub, including a new quasiperiodic full annular rub motion that is different from the traditional full annular rubs mentioned above.

The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system.

Double-scroll attractors are one of the pillars of modern chaos theory. However, rigorous computer-free analysis of their existence and global structure is often elusive. Here, we address this fundamental problem by constructing an analytically tractable piecewise-smooth system with a double-scroll attractor. We derive a Poincaré return map to prove the existence of the double-scroll attractor and explicitly characterize its global dynamical properties. In particular, we reveal a hidden set of countably many saddle orbits associated with infinite-period Smale horseshoes. These complex hyperbolic sets emerge from an ordered iterative process that yields sequential intersections between different horseshoes and their preimages. This novel distinctive feature differs from the classical Smale horseshoes, directly intersecting with their own preimages. Our global analysis suggests that the structure of the classical Chua attractor and other figure-eight attractors might be more complex than previously thought.

Bifurcation of Limit Cycles by Perturbing Piecewise Linear Hamiltonian Systems with Piecewise Polynomials

In this paper, we study a class of piecewise smooth near-Hamiltonian systems with piecewise polynomial perturbations. We first give the expression of the first order Melnikov function, and then estimate the number of limit cycles bifurcated from periodic annuluses by Melnikov function method. In addition, we discuss the number of limit cycles that can appear simultaneously near both sides of a generalized homoclinic or generalized double homoclinic loop.

Evaluating route to impact convergence of the harmonic balance method for piecewise-smooth systems

In this work, we investigate the applicability of the harmonic balance method (HBM) to predict periodic solutions of a single degree-of-freedom forced Duffing oscillator with freeplay nonlinearity. By studying the route to impact, which refers to a parametric study as the contact stiffness increases from soft to hard, the convergence behavior of the HBM can be understood in terms of the strength of the non-smooth forcing term. HBM results are compared to time-integration results to facilitate an evaluation of the accuracy of nonlinear periodic responses. An additional contribution of this study is to perform convergence and stability analysis specifically for isolas generated by the non-smooth nonlinearity. Residual error analysis is used to determine the approximate number of harmonics required to get results accurate to a given error tolerance. Hill’s method and Floquet theory are employed to compute the stability of periodic solutions and identify the types of bifurcations in the system.

Master stability function for piecewise smooth Filippov networks

We consider a network of identical piecewise smooth bimodal systems, also known as systems of Filippov type, that synchronizes along the asymptotically stable periodic orbit of a single agent. We explicitly characterize the fundamental matrix solution of the network along the synchronous solution and extend the Master Stability Function tool to the present case of non-smooth dynamics of Filippov type.

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.

A Comparison Study of Time-Domain Computation Methods for Piecewise Smooth Fractional-Order Circuit Systems

The role of fractional calculus in circuit systems has received increased attention in recent years. In order to evaluate the effectiveness of time-domain calculation methods in the analysis of fractional-order piecewise smooth circuit systems, an experimental prototype is developed, and the effects of three typical calculation methods in different test scenarios are compared and studied in this paper. It is proved that Oustaloup’s rational approximation method usually overestimates the peak-to-peak current and brings in the pulse–voltage phenomenon in piecewise smooth test scenarios, while the results of the two iterative recurrence-form numerical methods are in good agreement with the experimental results. The study results are dedicated to provide a reference for efficiently deploying calculation methods in fractional-order piecewise smooth circuit systems. Some quantitative analysis results are concluded in this paper.

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.

Coexistence of three heteroclinic cycles and chaos analyses for a class of 3D piecewise affine systems.

Our objective is to investigate the innovative dynamics of piecewise smooth systems with multiple discontinuous switching manifolds. This paper establishes the coexistence of heteroclinic cycles in a class of 3D piecewise affine systems with three switching manifolds through rigorous mathematical analysis. By constructing suitable Poincaré maps adjacent to heteroclinic cycles, we demonstrate the occurrence of two distinct types of horseshoes and show the conditions for the presence of chaotic invariant sets. A family of attractors that satisfy the criteria are presented using this technique. It is shown that the outcomes of numerical simulation accurately reflect those of our theoretical results.

Degenerate T-singularity bifurcation and crossing periodic orbits in a 3-dimensional piecewise smooth system

In this paper we investigate the dynamics of a 3-dimensional piecewise smooth system, which is an unfolding of a degenerate T-singularity. In local aspect, under the same non-degenerate condition given in [6] and determined by the linear part of perturbation, we analyze the bifurcation of this degenerate T-singularity on the switching manifold and give the bifurcation diagram including the degenerate two-fold bifurcation curve and the transcritical bifurcation curve. When this condition is not satisfied, a further non-degenerate condition determined by terms no more than degree two of perturbation is constructed, under which we obtain the pseudo-equilibrium bifurcation curve besides previous two bifurcation curves. In non-local aspect, for a system satisfying the second non-degenerate condition we prove that there exist either one family or two families of non-isolated crossing periodic orbits or at most two crossing periodic orbits and it is reachable. Corresponding necessary and sufficient conditions are given as well as periods and locations of these crossing periodic orbits.

Global stability of a Lotka-Volterra piecewise-smooth system with harvesting actions and two predators competing for one prey

The global dynamics of a three-dimensional Lotka-Volterra system described by two predator species competing for one prey and with human harvesting action on the predator species is studied in this paper. The harvesting action is introduced by means of two switching control actions defined on the predator species. A well-known result in the study of ecosystem modeling is that there are two states of coexistence of one of the predatory species with the prey species assuming that the principle of competitive exclusion or coexistence of competing species is fulfilled. In this sense, the three species cannot coexist in this class of system. In this work, it is proved that there is a global stable equilibrium point where the three species can coexist due to the proposed harvesting action.

Nonlinear Sliding and Nonlinear Regularization of Piecewise Smooth System

Nonlinear Sliding and Nonlinear Regularization of Piecewise Smooth System

A Filippov type predator prey system with selective harvesting of predator

In this paper, piece-wise smooth predator prey Filippov system to describe the role of refuge to prey and additional food to predators is explored. Logistic growth of prey and Holling type II predation is considered. Selective proportional harvesting of predators is included. Refuge is provided to a fraction of preys when their density is below a threshold value. This reduces the food availability for predators. To compensate this, additional food is provided to predators. The dynamics on the boundary is computed using the Filippov regularization method. The stability and existence of different equilibrium states of the independent smooth subsystems are analyzed. The role of quality of additional food with respect to prey in the characterization of the equilibrium points as regular or virtual equilibrium points for the Filippov system is investigated. It is verified that if only refuge is provided to prey with no additional food to predators then both the interior equilibrium state will never be regular simultaneously for the Filippov system. It is observed from the sliding mode dynamics that there is no escaping region. It is proved that both refuge to prey and additional food to predators enhances the sliding region on the boundary. The analysis of boundary equilibrium points, tangent points and pseudoequilibrium points on the boundary is carried out. The role of effort on harvesting on the expansion or shrinking of the limit cycle is elaborated numerically. Boundary bifurcation for the interior equilibrium state of the subsystems is demonstrated. Numerical simulation is used to elaborate the dynamics of Filippov system.

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.