Piecewise Smooth Systems Research Articles

On the maximum number of limit cycles for a piecewise smooth differential system

In this paper, we consider an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of x¨+x=0 under piecewise smooth perturbation. By using the first order average method, we prove that the maximum number is at most n and n can be reached.

Global analysis of the dynamics of a mathematical model to intermittent HIV treatment

The aim of this paper is to study the qualitative dynamics of a piecewise smooth system modeling the intermittent treatment of the human immunodeficiency virus. Typical singularities and closed orbits are observable, and we quantitatively explore the dynamics around those singularities and closed orbits. Moreover, we conclude that this protocol always will be successful since the trajectory passing through any initial condition converges to one of these distinguished orbits. Our formal mathematical results corroborate the real-world observation, where the virus is not eliminated, but the number of infected cells is controlled around a specific value.

Multiple boundaries sliding mode control applied to capacitor voltage-balancing systems

Capacitor voltage-balancing systems are usually applied to power electronic circuits. The main issue in these systems is equalising the voltage of a large number of capacitors connected to a voltage source in serial or parallel arrangements. A particular application of the capacitor voltage-balancing systems is found in modular multilevel converters (MMC). The voltage balancing of the floating capacitors in the submodules of the MMC during its pre-charging operating stage is a key issue since it is a critical task for the correct operating of these converters. The stability analysis of an active voltage balancing strategy for pre-charging MMCs is addressed in this paper. The adopted voltage-balancing strategy consists in adding a resistance to each submodule of the MMC by means of a controlled switch. These switches are being controlled by a sliding mode control algorithm with multiple boundaries (discontinuity surfaces of high co-dimension). These systems are essentially discontinuous piecewise smooth dynamical systems (Filippov systems) commanded by electronic switches. In this paper, the local stability of the voltage balanced system is analytically proven for an arbitrary number of submodules. In addition, a detailed analysis of the global dynamics of this system with two submodules and two switching boundaries sliding mode control is presented. Simulation results obtained on a MMC with 10 submodules are shown to validate the theoretical analysis.

The Regularized Visible Fold Revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $$\epsilon \rightarrow 0$$ . Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $$\epsilon \rightarrow 0$$ , grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law.

Sliding Shilnikov connection in Filippov-type predator–prey model

Recently, a piecewise smooth differential system was derived as a model of a 1 predator-2 prey interaction where the predator feeds adaptively on its preferred prey and an alternative prey. In such a model, strong evidence of chaotic behavior was numerically found. Here, we revisit this model and prove the existence of a Shilnikov sliding connection when the parameters are taken in a codimension one submanifold of the parameter space. As a consequence of this connection, we conclude, analytically, that the model behaves chaotically for an open region of the parameter space.

Bursting oscillations with delayed C-bifurcations in a modified Chua’s circuit

In this research, a typical Chua’s circuit with a piecewise nonlinear resistor and a slow-varying periodic excitation is considered to investigate the dynamical mechanisms of bursting oscillations in the piecewise-smooth dynamical system. A set of new bursting oscillations is observed when the amplitude of the excitation is changed. By regarding the excitation term as a bifurcation parameter, the codimension-1 conventional bifurcations and non-smooth bifurcations of the fast subsystem are explored. Fold bifurcation, supercritical Hopf (sup-Hopf) bifurcation, non-smooth Hopf bifurcation, grazing bifurcation, and C-bifurcation are discovered via theoretical and numerical methods. The C-bifurcation connects the stable limited cycle bifurcated from non-smooth Hopf bifurcation with the stable limited cycle bifurcated from the sup-Hopf bifurcation. When the fast subsystem driven by the slow subsystem passes through the bifurcation points, slow passage effect near the non-smooth Hopf bifurcation and delay of the C-bifurcation take place. The delayed C-bifurcation may lead to multiple transition patterns between different attractors, including two transition patterns of reverse direction near the fold and sup-Hopf bifurcations. The delayed transition to other attractor creates a non-smooth hysteresis loop and enables the generation of bursting oscillations.

Dynamics of a driven harmonic oscillator coupled to independent Ising spins in random fields.

We aim at an understanding of the dynamical properties of a periodically driven damped harmonic oscillator coupled to a Random Field Ising Model (RFIM) at zero temperature, which is capable of showing complex hysteresis. The system is a combination of a continuous (harmonic oscillator) and a discrete (RFIM) subsystem, which classifies it as a hybrid system. In this paper we focus on the hybrid nature of the system and consider only independent spins in quenched random local fields, which can already lead to complex dynamics such as chaos and multistability. We study the dynamic behavior of this system by using the theory of piecewise-smooth dynamical systems and discontinuity mappings. Specifically, we present bifurcation diagrams and Lyapunov exponents as well as results for the shape and the dimensions of the attractors and the self-averaging behavior of the attractor dimensions and the magnetization. Furthermore we investigate the dynamical behavior of the system for an increasing number of spins and the transition to the thermodynamic limit, where the system behaves like a driven harmonic oscillator with an additional nonlinear smooth external force.

Hopf bifurcation of limit cycles by perturbing piecewise integrable systems

In this paper, we study a class of piecewise smooth near-integrable non-Hamilton systems. By using the first order Melnikov function, we obtain the maximal number of limit cycles in Hopf bifurcation under a class of perturbations.

Mathematical model of an antiretroviral therapy to HIV via Filippov theory

This work aims to study some qualitative features of piecewise smooth systems applied to the dynamics of the Human Immunodeficiency Virus (HIV). The application of sliding mode control and strategies of structured treatment interruptions in HIV is a new and challenging research area, which deserves attention and improvement. We investigate the dynamic behavior of the discontinuous mathematical model and we observe the occurrence of typical singularities.

Piecewise-Smooth Slow–Fast Systems

We deal with piecewise-smooth differential systems $\dot {z}=X(z), z=(x,y)\in \mathbb {R}\times \mathbb {R}^{n-1},$ with switching occurring in a codimension one smooth surface Σ. A regularization of X is a 1-parameter family of smooth vector fields Xδ,δ > 0, satisfying that Xδ converges pointwise to X in $\mathbb {R}^{n}\setminus {\Sigma }$ , when $\delta \rightarrow 0$ . The regularized system $\dot {z}=X^{\delta }(z)$ is a slow–fast system. We work with two known regularizations: the classical one proposed by Sotomayor and Teixeira and its generalization, using transition functions without imposing the monotonicity condition. Minimal sets of regularized systems are studied with tools of the geometric singular perturbation theory. Moreover, we analyzed the persistence of the sliding region of piecewise-smooth slow–fast systems by singular perturbations.

Nonlinear Aeroelastic Responses of an Airfoil with a Control Surface by Precise Integration Method

This paper presents a highly accurate algorithm, based on precise integration method (PIM), to investigate the aeroelastic system of an airfoil with control surface. The nonlinear system is separated as linear sub-systems, which are solved by PIM stepwise. During each step, a predictor-corrector algorithm is proposed to detect switching points between sub-systems. Numerical examples show that, the presented algorithm is much more accurate and efficient than the widely-used Runge-Kutta method. Sometimes, the Runge-Kutta method even provides false results. With such high precision and efficiency, the presented algorithm has the potential to become a benchmark for comparison in solving piecewise-smooth dynamical systems.

Discontinuity mappings for stochastic nonsmooth systems

For stability and bifurcation analysis involving recurrent behaviour such as periodic orbits, it is important to be able to quantify how nearby trajectories behave by means of a local mapping. In smooth systems these mappings can be computed using the system’s variational equations. For piecewise-smooth or hybrid systems the same technique cannot be used without some corrections. This is due to the fact that nearby trajectories can be topologically distinct because they can undergo different sequences of events associated with the system’s discontinuity boundaries. To account for this, one can derive zero-time discontinuity mappings associated with boundary interactions. In this paper we derive zero-time discontinuity mappings for piecewise-smooth vector fields and hybrid dynamical systems in which the position of the discontinuity boundary has a stochastic component. In particular, we consider systems with stochastically oscillating boundaries and systems with stochastic imperfections on the discontinuity boundary.

Limit cycles bifurcating from piecewise quadratic systems separated by a straight line

In this paper, we are concerned with limit cycles bifurcating from piecewise quadratic systems separated by a straight line. By means of the first-order averaging theory of discontinuous differential systems and the Argument Principle, we present an estimate of the maximum number of limit cycles bifurcating from the period annulus around the center of piecewise smooth systems.

On the Study and Application of Limit Cycles of a Kind of Piecewise Smooth Equation

In this paper we devote to study the piecewise smooth equation of the form: $$\begin{aligned} \frac{dx}{dt}=S(t,x)=\left\{ \begin{array}{ll} S_1(t,x)=a_1(t)x^m+b(t), &{} \quad \hbox {if }\quad x\ge {0}, \\ S_2(t,x)=a_2(t)x^m+b(t), &{} \quad \hbox {if }\quad x 0$$). In this study we pay more attention to the examples in which the equation has limit cycle(s) crossing the separation straight line $$x=0$$. In the end, we apply this result on a kind of piecewise smooth planar system which has a separation curve $$x^2+y^2=1$$.

Estimating the dynamics of systems with noisy boundaries

In a smooth dynamical system the characteristics of a given reference trajectory can be determined, to lowest order, by examining the linearised system about the reference trajectory. In other words, we can approximate the deviations of trajectories after a given time, with starting points in a neighbourhood of the reference trajectory, by multiplying the initial deviations by the corresponding fundamental matrix solution.This form of analysis cannot be used directly in nonsmooth systems as the vector field is either not everywhere differentiable or the flow function is not continuous. To account for this, one can derive the zero-time discontinuity mapping (ZDM) associated with the discontinuity boundary. The Jacobian of this mapping is known as the saltation matrix and its properties can tell us how the crossing of the discontinuity boundary affects the deviations of trajectories from a reference trajectory. In particular, this matrix can be composed with the fundamental matrix solutions of the individual flows on either side of the discontinuity boundary in order to determine the overall fundamental matrix solution of a trajectory that crosses the boundary.In this paper we derive a saltation matrix for a piecewise-smooth dynamical system in which the position of the discontinuity boundary oscillates according to a mean-reverting stochastic process. The derived saltation matrix contains the entire effect of both the discontinuity and the uncertainty introduced into the system by the noisy boundary, and is composable with the deterministic fundamental matrix solutions of the individual flows to give the overall fundamental matrix solution of a crossing trajectory.We also present some simple examples of piecewise-smooth systems with stochastically varying boundaries, analysed using the derived noisy saltation matrix. In particular we focus on the analysis of a discontinuous variant of the Chua circuit. In this case we apply noise to the system’s discontinuity boundaries which are generated by the piecewise-linear nature of the voltage–current response of the Chua diode. We find that our method allows us to analyse the effects of boundary noise on periodic attractors close to bifurcation points. In particular we show that we can use the method to accurately predict the noise amplitudes required to destroy or merge periodic attractors.

Multistability in a quasiperiodically forced piecewise smooth dynamical system

Considering a class of quasiperiodically forced piecewise smooth systems, we uncover a dynamic phenomenon in which strange nonchaotic attractors (SNAs) and quasiperiodic attractors coexist in nonsmooth dynamical system, obtaining the domains of attraction of these coexisting attractors in parameter space in order to analyze the global dynamics. The global dynamics analysis demonstrates that SNAs are the transition from quasiperiodic attractors to chaotic attractors. The routes to SNAs, including torus-doubling route, torus fractalization route or, simply, fractal route, and intermittency route, are also investigated. The characteristics of SNAs are described by dynamical invariants such as the Lyapunov exponent, power spectrum, phase sensitivity and rational approximations.

FURTHER STUDIES ON LIMIT CYCLE BIFURCATIONS FOR PIECEWISE SMOOTH NEAR-HAMILTONIAN SYSTEMS WITH MULTIPLE PARAMETERS<inline-formula><tex-math id="M1">$ ^* $</tex-math></inline-formula>

This paper investigates the limit cycle bifurcations for piecewise smooth near-Hamiltonian systems with multiple parameters. The formulas for the second and third term in expansions of the first order Melnikov function are derived respectively. The main results improve some known conclusions.

Study of Periodic Orbits in Periodic Perturbations of Planar Reversible Filippov Systems Having a Twofold Cycle

We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed system has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed.

A note on attractivity for the intersection of two discontinuity manifolds

In piecewise smooth dynamical systems, a co-dimension 2 discontinuity manifold can be attractive either through partial sliding or by spiraling. In this work we prove that both attractivity regimes can be analyzed by means of the moments solution, a spiraling bifurcation parameter and a novel attractivity parameter, which changes sign when attractivity switches from sliding to spiraling attractivity or vice-versa. We also study what happens at what we call attractivity transition points, showing that the spiraling bifurcation parameter is always zero at those points.

Non-observable chaos in piecewise smooth systems

In the present paper, we discuss bifurcations of chaotic attractors in piecewise smooth one-dimensional maps with a high number of switching manifolds. As an example, we consider models of DC/AC power electronic converters (inverters). We demonstrate that chaotic attractors in the considered class of models may contain parts of a very low density, which are unlikely to be observed, neither in physical experiments nor in numerical simulations. We explain how the usual bifurcations of chaotic attractors (merging, expansion and final bifurcations) in piecewise smooth maps with a high number of switching manifolds occur in a specific way, involving low-density parts of attractors, and how this leads to an unusual shape of the bifurcation diagrams.