Piecewise Smooth Systems Research Articles

Bursting oscillations with non-smooth Hopf-fold bifurcations of boundary equilibria in a piecewise-smooth system

Bursting oscillations with non-smooth Hopf-fold bifurcations of boundary equilibria in a piecewise-smooth system

Hopf Cyclicity and Center Conditions for Some Piecewise Smooth Cubic Differential Systems

In this paper, we consider piecewise smooth differential systems with subsystems having cubic reversible isochronous centers at the origin, where the separation line is a straight line passing through the origin. By computing Lyapunov constants, we obtain Hopf cyclicity at the origin with necessary and sufficient conditions for the origin to be a center in this type of piecewise smooth differential systems.

Grazing-sliding bifurcation in a dry-friction oscillator on a moving belt under periodic excitation.

In this paper, we consider the grazing-sliding bifurcations in a dry-friction oscillator on a moving belt under periodic excitation. The system is a nonlinear piecewise smooth system defined in two zones whose analytical expressions of the solutions are not available. Thus, we obtain conditions of the existence of grazing-sliding orbits numerically by the shooting method. Then, we compute the lower and higher order approximations of the stroboscopic Poincaré map, respectively, near the grazing-sliding bifurcation point by the method of local zero-time discontinuity mapping. The results of computing the bifurcation diagrams obtained by the lower and higher order maps, respectively, are compared with those from direct simulations of the original system. We find that there are big differences between the lower order map and the original system, while the higher order map can effectively reduce such disagreements. By using the higher order map and numerical simulations, we find that the system undergoes very complicated dynamical behaviors near the grazing-sliding bifurcation point, such as period-adding cascades and chaos.

Topology Identification for Networked Piecewise-smooth Systems With Multiple Weight Couplings Based a Novel Fixed-time Synchronization Approach

Topology Identification for Networked Piecewise-smooth Systems With Multiple Weight Couplings Based a Novel Fixed-time Synchronization Approach

Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status

Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status

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.

Canard cycle and nonsmooth bifurcation in a piecewise-smooth continuous predator-prey model

This article establishes a bifurcation analysis of a singularly perturbed piecewise-smooth continuous predator–prey system with a sufficiently small parameter. The bifurcation that can generate limit cycles here is our main concern. To achieve this goal, we have developed a lemma that is used to determine the parameter region that can generate limit cycles. Further conclusions indicate that the existence of a 2-shaped critical manifold is required. Based on the Poincaré-Bendixon lemma, Fenichel’s theory and geometric singular perturbation theory, we demonstrate the possibility of generating smooth and nonsmooth bifurcations. In fact, nonsmooth bifurcations only occur in piecewise-smooth systems. More specifically, different types of nonsmooth bifurcations are also presented in this article, including nonsmooth Hopf bifurcation, Hopf-like bifurcation and super-explosion. In addition, this article discusses the existence of crossing limit cycles and explains whether the crossing limit cycle is characterized by canard cycles without head, canard cycles with head or relaxation oscillations. Furthermore, the coexistence of two relaxation oscillations, the coexistence of two canard cycles without head, and the coexistence of one relaxation oscillation and one canard cycle without head are investigated. Moreover, the one-parameter bifurcation diagram is also presented in this paper through numerical simulations.

Limit Cycles Bifurcating from Planar Polynomial Quasi-Homogeneous Centers of Weight-Degree 3 with Nonsmooth Perturbations

In this paper, we estimate the number of limit cycles bifurcating from the periodic orbits of the weight-homogeneous polynomial centers of weight-degree 3, when they are perturbed inside the class of piecewise smooth polynomial systems of degree n. By using the first-order averaging method, we give the upper and lower bounds on the maximal number of limit cycles which can bifurcate from the period annuluses. Our results indicate that the nonsmooth systems can have more limit cycles than the smooth ones.

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.

К выбору параметров регулятора непосредственного понижающего преобразователя с одноконтурной системой управления с учетом динамических нелинейностей

The article is devoted to the issues of choosing the parameters of a proportional-integral-differentiating regulator of a single loop control system with a buck converter taking into account dynamic nonlinearities This problem arises when it is necessary to design pulsed voltage conver¬ters with increased performance, when it is necessary to increase the cutoff frequency of an open-loop control loop without the possibility of increasing the carrier frequency of pulse width modulation due to limitations of the element base, which in some cases leads to the appearance of nonlinear effects. This must be taken into account when choosing both the open-loop cutoff frequency and the stability margin. The article proposes a nonlinear dynamic model of a direct buck converter with a proportional-integral-differentiating regulator both in the form of a piecewise smooth system of differential equations and in the form of a Poincaré map. A method for selecting controller parameters is proposed for a given open-loop cutoff frequency, and a given stability margin, as well as for given ranges of input voltage and load resistance based on the use of numerical methods for solving systems of nonlinear equations with given restrictions. The nonlinear dynamics of the system was studied for various sets of controller parameters. It is shown that with an increase in the cutoff frequency and small stability margins, undesirable dynamic modes may arise in the system when the filter capacitance drops within the calculated values, while linear dynamic models show the stability of the system. To eliminate the possibility of the emergence of undesirable modes, it is necessary to increase the stability margin with subsequent monitoring of the results using a nonlinear dynamic model. The proposed method for selecting the cutoff frequency of an open loop and the phase stability margin, based on the analysis of nonlinear dyna¬mics, allows you to design power supplies with increased performance.

Limit cycles of piecewise smooth differential systems of the type nonlinear centre and saddle

Piecewise linear differential systems separated by two parallel straight lines of the type of centre-centre-Hamiltonian saddle and the centre-Hamiltonian saddle-Hamiltonian saddle can have at most one limit cycle and there are systems in these classes having one limit cycle. In this paper, we study the limit cycles of a piecewise smooth differential system separated by two parallel straight lines formed by nonlinear centres and a Hamiltonian saddle.

Chaos detection and control of a fractional piecewise-smooth system with nonlinear damping

Chaotic response is a robust effect in natural systems, and it is usually unfavorable for applications owing to uncertainty. In this paper, we propose several control strategies to stabilize the chaotic rhythm of a fractional piecewise-smooth oscillator. First, the Melnikov analysis is applied to the system, and the critical condition for the occurrence of homoclinic chaos is scrupulously established. Then, by applying appropriate control mechanisms, including delayed feedback control and periodic excitations, to the system, we can eliminate the zeros in the original Melnikov function, which serve as sufficient criteria for chaos suppression. Numerical simulations further demonstrate the accuracy of the theoretical results and the validity of the control schemes. Finally, the effects of parameter variations on the efficiency of control strategies are investigated. Note that we use the complex Simpson formula to calculate the complicated Melnikov functions presented in this paper. The current work may open a new innovative path to detect and control the chaotic dynamics of fractional non-smooth models.

Homoclinic and heteroclinic bifurcations in piecewise smooth systems via stability-changing method

Homoclinic and heteroclinic bifurcations in piecewise smooth systems via stability-changing method

Invariant tori, topological horseshoes, and their coexistence in piecewise smooth hybrid systems

Invariant tori, topological horseshoes, and their coexistence in piecewise smooth hybrid systems

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.

Non-isochronicity on piecewise Hamiltonian differential systems with homogeneous nonlinearities

In this paper, we prove the non-isochronicity of Σ-centres for a class of planar piecewise smooth differential systems with a straight switching line, whose two sub-systems are Hamiltonian differential systems with a non-degenerated centre and only homogeneous nonlinearities. This result generalizes a classical result of non-isochronicity on the smooth Hamiltonian systems with homogeneous nonlinearities.

Strange nonchaotic attractors in a class of quasiperiodically forced piecewise smooth systems

Strange nonchaotic attractors in a class of quasiperiodically forced piecewise smooth systems

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

Limit cycle oscillation and dynamical scenarios in piecewise-smooth nonlinear systems with two-sided constraints

Limit cycle oscillation and dynamical scenarios in piecewise-smooth nonlinear systems with two-sided constraints