Piecewise Smooth Systems Research Articles

Piecewise parametric chaotic model of p53 network based on the identified unifying framework of divergent p53 dynamics

Being the core of the large p53 protein (cancer) network, the ATM-p53-Wip1 sub-network is known to confer monostable oscillations and fixed levels of p53 to the p53 network. Using a 2-Dimensional parametric model of this sub-network proposed by [68], we seek to know what other dynamical patterns are possible. Not to miss out on any pattern, we analytically identify disjoint behavioral regions whose union covers the entire domain of equilibria in phase space. There exist five qualitatively different phase portraits leading to monostable dynamics, birhythmicity, single pulse dynamics and, typical and atypical bistability. Such patterns are known to exist in the p53 network in wet-lab settings, which evokes the idea that the underlying structure might be the ATM-p53-Wip1 sub-network. Additionally, the extension of the model by an accumulating variable, modeling the pro-apoptotic components of the p53 network, has the potential to construct a homoclinic orbit, which is shown in a piecewise fashion. Then, we propose a piecewise system by explicitly embedding the perceived HC orbit using a biologically meaningful sliding surface. Using the Shilnikov Theorem for piecewise smooth systems, we demonstrate mathematically the previously unobserved possibility of a chaotic mechanism in the p53 network, which awaits further biological confirmation. Chaos is observed under the stringent and pathological conditions of failed-apoptosis and unrepairable DNA damage. Thus, our results support the hypothesis that chaotic gene expressions accompany tumor formation.

Modeling the influence of health education on individual behavior in the process of disease transmission

Health education directly affects the different behaviors of individuals who visit and do not visit after illness by reporting the number of patients and their changing rate. As a result, visiting and non‐visiting patients have different infection rates, cure rates, and disease‐related mortality. In response to this impact, this article establishes a model to further analyze the impact of health education on the spread of diseases. We divide patients into visiting and non‐visiting patients and introduce a weighting function that depends on the number of visiting patients and their changing rate. We constructs an implicit differential equations and use the properties of Lambert W function and its inverse function to transform it into piecewise smooth systems with explicit definitions. According to the analysis method of switched system, the two subsystems are analyzed qualitatively. The results show that health education cannot affect the basic reproduction number of disease transmission, but it causes the system to produce many types of equilibria. In addition, the difference in the healing rate and self‐healing rate between visiting and non‐visiting patients will affect the number of visiting and non‐visiting patients and the stability of the equilibria. Moreover, the system will generate fold branch. In addition, the patient's healing/self‐healing rate and the implementation of health education will affect the appearance of virtual or false equilibrium. Numerical simulation and sensitivity analysis further confirm that the total number of visiting and non‐visiting patients will decrease with the increase of health education. Finally, our research results show that health education can reduce the infection rate only if there are enough susceptible people receiving health education.

Correction to: The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a Switching Manifold

Correction to: The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a Switching Manifold

Global Dynamics of a Piecewise Smooth System with a Fold–Cusp and General Parameters

Global Dynamics of a Piecewise Smooth System with a Fold–Cusp and General Parameters

On a High-Precision Method for Studying Attractors of Dynamical Systems and Systems of Explosive Type

The author of this article considers a numerical method that uses high-precision calculations to construct approximations to attractors of dynamical systems of chaotic type with a quadratic right-hand side, as well as to find the vertical asymptotes of solutions of systems of explosive type. A special case of such systems is the population explosion model. A theorem on the existence of asymptotes is proved. The extension of the numerical method for piecewise smooth systems is described using the Chua system as an example, as well as systems with hysteresis.

Bifurcation of limit cycles from a parabolic–parabolic type critical point in a class of planar piecewise smooth quadratic systems

In this paper we study small amplitude limit cycles bifurcated from a parabolic–parabolic (PP) type critical point of planar piecewise smooth quadratic systems having exactly one switching line given by the x-axis. Besides the non-smoothness, more difficulties arise when the critical point has a parabolic contact. For such kind of critical point, to make sure that the return map is analytic, one has to use the generalized polar coordinates instead of the classic polar coordinates to compute the corresponding Lyapunov constants. Consequently, much more complicated computations are involved. By the recent results of Novaes and Silva, the index of the first nonzero Lyapunov constant for a PP type critical point is always an even number 2ℓ+2 with ℓ≥1. We call the corresponding weak focus (0,0) is of order ℓ. We obtain eight center conditions and six conditions under which (0,0) is a weak focus of order 6. Furthermore, we prove that at least seven limit cycles can bifurcate from (0,0).

Local cyclicity and criticality in FF-type piecewise smooth cubic and quartic Kukles systems

We investigate small-amplitude crossing limit cycles and local critical periods for some classes of piecewise smooth Kukles systems with a focus–focus type (abbr.FF-type) equilibrium point, which is formed by two regions separated by a straight line. In this class, we acquire new lower bounds for the local cyclicity Mpff(n) and the criticality Cp(n) with n=3,4. More precisely, Mpff(3)≥10, Mpff(4)≥12 and Cp(3)≥6 for the discontinuous case without a sliding segment, and Mpff(3)≥9, Mpff(4)≥13 and Cp(3)≥6 for the continuous case. Furthermore, one more limit cycle is obtained for the discontinuous case with a sliding segment in comparison with the one without a sliding segment, namely 11 for cubic system and 13 for quartic one. These lower bounds are also new, that are known up to now, for the general FF-type piecewise smooth cubic and quartic systems around an equilibrium. Besides, we give the Finite Order Bifurcation Lemma of local critical periods in FF-type piecewise smooth systems, which is the generalization of the smooth ones.

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.

Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

In this paper, we study limit cycle bifurcations for planar piecewise smooth near-Hamiltonian systems with nth-order polynomial perturbation. The piecewise smooth linear differential systems with two centers formed in two ways, one is that a center-fold point at the origin, the other is a center-fold at the origin and another unique center point exists. We first explore the expression of the first order Melnikov function. Then by using the Melnikov function method, we give estimations of the number of limit cycles bifurcating from the period annulus. For the latter case, the simultaneous occurrence of limit cycles near both sides of the homoclinic loop is partially addressed.

Bifurcations of a Generalized Heteroclinic Loop in a Planar Piecewise Smooth System with Periodic Perturbations

Bifurcations of a Generalized Heteroclinic Loop in a Planar Piecewise Smooth System with Periodic Perturbations

Melnikov functions of arbitrary order for piecewise smooth differential systems in [formula omitted] and applications

In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in Rn with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n-dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed.

NOSNOC: A Software Package for Numerical Optimal Control of Nonsmooth Systems

This letter introduces the NOnSmooth Numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) with piecewise smooth systems (PSS). The tool supports: 1) automatic reformulation of systems with state jumps into PSS (via the time-freezing reformulation 1) and of PSS into computationally more convenient forms, 2) automatic discretization of the OCP via, e.g., the recently introduced Finite Elements with Switch Detection 2 which enables high accuracy optimal control and simulation of PSS, 3) solution methods for the resulting discrete-time OCP. The nonsmooth discrete-time OCP are solved with techniques of continuous optimization in a homotopy procedure, without the use of integer variables. This enables the treatment of a broad class of nonsmooth systems in a unified way. Two tutorial examples are given. A benchmark shows that NOSNOC provides both faster and more accurate solutions than conventional approaches, including mixed-integer formulations.

Continuous Optimization for Control of Hybrid Systems With Hysteresis via Time-Freezing

This article regards numerical optimal control of a class of hybrid systems with hysteresis using solely techniques from nonlinear optimization, without any integer variables. Hysteresis is a rate independent memory effect which often results in severe nonsmoothness in the dynamics. These systems are not simply Piecewise Smooth Systems (PSS); they are a more complicated form of hybrid systems. We introduce a time-freezing reformulation which transforms these systems into a PSS. From the theoretical side, this reformulation opens the door to study systems with hysteresis via the rich tools developed for Filippov systems. From the practical side, it enables the use of the recently developed Finite Elements with Switch Detection [Nurkanovic et al., 2022], which makes high accuracy numerical optimal control of hybrid systems with hysteresis possible. We provide a time optimal control problem example and compare our approach to mixed-integer formulations from the literature.

Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation

In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\dot{X}=\begin{cases} (H_y^+,-H_x^+) & \mbox{if}~y>\varepsilon (H_y^-,-H_x^-) & \mbox{if}~y<\varepsilon f(x) \end{cases}$ equals to the number of positive zeros of $f$ when at $\varepsilon=0$ the system has a period annulus around the origin.

Border Collision Bifurcations and Chaos in a Ring of Three Unidirectionally Coupled Nonmonotonic Piecewise Constant Neurons

The bifurcations and chaos in a ring of three nonmonotonic piecewise constant neurons with unidirectional inhibitory coupling are examined. Periodic solutions in the system undergo border collision bifurcations, which are characteristic of piecewise smooth systems. Conditions for the bifurcations of periodic solutions are derived analytically by using a piecewise constant function as the output function of neurons. Multiple periodic solutions are generated through border collision bifurcations and saddle-node bifurcations. Chaotic attractors are generated through border collision bifurcations and degenerate period-doubling bifurcations. The existence of multiple periodic solutions and chaotic attractors is proven to be common in rings of unidirectionally coupled nonmonotonic neurons.

Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones

&lt;p style='text-indent:20px;'&gt;In this paper, we consider the nonsmooth bifurcation around a class of critical crossing cycles, which are codimension-2 closed orbits composed of tangency singularities and regular orbits, for a two-parameter family of planar piecewise smooth system with two zones. By the construction of suitable displacement function (equivalently, Poincar&lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ {\rm\acute{e}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; map), the stability and the existence of periodic solutions under the variation of the parameters inside this system are characterized. More precisely, we obtain some parameter regions on the existence of crossing cycles and sliding cycles near those loops. As applications, several examples are given to illustrate our main conclusions.&lt;/p&gt;

Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two

&lt;p style='text-indent:20px;'&gt;In this paper we consider an &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \Sigma $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; which is an intersection of two hyperplanes &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \Sigma_1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \Sigma_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. We investigate the relation between periodic orbit of PWS system and periodic orbit of its double regularized system. If this PWS system has an asymptotically stable sliding periodic orbit(including type Ⅰ and type Ⅱ), we establish conditions to ensure that also a double regularization of the given system has a unique, asymptotically stable, periodic orbit in a neighbourhood of &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \gamma $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, converging to &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \gamma $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; as both of the two regularization parameters go to &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ 0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; by applying implicit function theorem and geometric singular perturbation theory.&lt;/p&gt;

Chaotic threshold of a class of hybrid piecewise-smooth system by an impulsive effect via Melnikov-type function

&lt;p style='text-indent:20px;'&gt;In this paper, we study the chaotic behavior of a class of hybrid piecewise-smooth system incorporated into an impulsive effect (HPSS-IE) under a periodic perturbation. More precisely, we assume that the unperturbed system with a homoclinic orbit, it transversally jumps across the first switching manifold by an impulsive stimulation and continuously crosses the second switching manifold. Then the corresponding Melnikov-type function is derived. Based on the new Melnikov-type function, the bifurcation and chaotic threshold of the perturbed HPSS-IE are analyzed. Furthermore, numerical simulations are precisely demonstrated through a concrete example. The results indicate that it is an extension work of previous references.&lt;/p&gt;

Chains in 3D Filippov systems: A chaotic phenomenon

This work is devoted to the study of global connections between typical generic singularities, named T-singularities, in piecewise smooth dynamical systems. Such a singularity presents the so-called nonsmooth diabolo, which consists on a pair of invariant cones emanating from it.We analyze global features arising from the communication between the branches of a nonsmooth diabolo of a T-singularity and we prove that, under generic conditions, such communication leads to a chaotic behavior of the system. More specifically, we relate crossing orbits of a Filippov system presenting certain crossing self-connections to a T-singularity, with a Smale horseshoe of a first return map associated to the system. The techniques used in this work rely on the detection of transverse intersections between invariant manifolds of a hyperbolic fixed point of saddle type of such a first return map and the analysis of the Smale horseshoe associated to it.From the specific case discussed in our approach, we present a robust chaotic phenomenon for which its counterpart in the smooth case seems to happen only for highly degenerate systems.

Sliding Bifurcation in Planar Piecewise Smooth Systems with Two Zones Separated by an Ellipse

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.