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A mathematical model for a nonlinear oscillator, which is composed of an oscillating mass interacting with a freely sliding friction damper, is introduced and investigated. This oscillator is a strongly simplified model for a damping principle applied to turbine blades to suppress oscillations induced by inhomogeneous flow fields. It exhibits periodic, quasi-periodic, as well as chaotic dynamics occuring suddenly due to adding sliding bifurcations. Mathematically, the oscillator is given as a piecewise smooth (Filippov) system with a switching manifold corresponding to the sticking phase of the damper mass. The rich dynamics of this system is analyzed and illustrated by means of resonance curves, Lyapunov diagrams, Poincaré sections and reductions to iterated one-dimensional maps.

This paper proposes a Filippov epidemic model with piecewise continuous function to represent the enhanced vaccination strategy being triggered once the proportion of the susceptible individuals exceeds a threshold level. The sliding bifurcation and global dynamics for the proposed system are investigated. It is shown that as the threshold value varies, the proposed system can exhibit variable sliding mode domains and local sliding bifurcations including boundary node (focus) bifurcation, double tangency bifurcation and other sliding mode bifurcations. Model solutions ultimately approach either one of two endemic states for two structures or the pseudo-equilibrium on the switching surface, depending on the threshold level. The findings indicate that proper combinations of threshold level and enhanced vaccination rate based on threshold policy can lead disease prevalence to a previously chosen level if eradication of disease is impossible.

A Filippov system is proposed to describe the stage structured nonsmooth pest growth with threshold policy control (TPC). The TPC measure is represented by the total density of both juveniles and adults being chosen as an index for decisions on when to implement chemical control strategies. The proposed Filippov system can have three pieces of sliding segments and three pseudo-equilibria, which result in rich sliding mode bifurcations and local sliding bifurcations including boundary node (boundary focus, or boundary saddle) and tangency bifurcations. As the threshold density varies the model exhibits the interesting global sliding bifurcations sequentially: touching → buckling → crossing → sliding homoclinic orbit to a pseudo-saddle → crossing → touching bifurcations. In particular, bifurcation of a homoclinic orbit to a pseudo-saddle with a figure of eight shape, to a pseudo-saddle-node or to a standard saddle-node have been observed for some parameter sets. This implies that control outcomes are sensitive to the threshold level, and hence it is crucial to choose the threshold level to initiate control strategy. One more sliding segment (or pseudo-equilibrium) is induced by the total density of a population guided switching policy, compared to only the juvenile density guided policy, implying that this control policy is more effective in terms of preventing multiple pest outbreaks or causing the density of pests to stabilize at a desired level such as an economic threshold.

Global dynamics of a class of planar Filippov systems with symmetry, which is a discontinuous limit case of a smooth oscillator, is studied. Necessary and sufficient conditions for the existence and the number of limit cycles are given. It is shown that at most two limit cycles or a pair of grazing loops exist. A special method is introduced to study grazing bifurcation. The monotonicity and the $C^{\infty }$ smoothness of the grazing bifurcation curve are proved. All global phase portraits and a complete global bifurcation diagram are described. Finally, some numerical examples are demonstrated.

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

In order to reduce the spread of plant diseases and maintain the number of infected trees below an economic threshold, we choose the number of infected trees and the number of susceptible plants as the control indexes on whether to implement control strategies. Then a Filippov plant-disease model incorporating cutting off infected branches and replanting susceptible trees is proposed. Based on the theory of Filippov system, the sliding mode dynamics and conditions for the existence of all the possible equilibria and Lotka–Volterra cycles are presented. We find that model solutions ultimately approach the positive equilibrium that lies in the region above the infected threshold value $T_I$, or the periodic trajectories that lie in the region below $T_I$, or the pseudo-attractor $E_T=(T_S,T_I)$, as we vary the susceptible and infected threshold values. It indicates that the plant-disease transmission is tolerable if the trajectories approach $E_T=(T_S,T_I)$ or the periodic trajectories lie in the region below $T_I$. Hence an acceptable level of the number of infected trees can be achieved when the susceptible and infected threshold values are chosen appropriately.

A multiscale system for environmentally-driven infectious disease is proposed, in which control measures at three different scales are implemented when the number of infected hosts exceeds a certain threshold. Our coupled model successfully describes the feedback mechanisms of between-host dynamics on within-host dynamics by employing one-scale variable guided enhancement of interventions on other scales. The modeling approach provides a novel idea of how to link the large-scale dynamics to small-scale dynamics. The dynamic behaviors of the multiscale system on two time-scales, i.e. fast system and slow system, are investigated. The slow system is further simplified to a two-dimensional Filippov system. For the Filippov system, we study the dynamics of its two subsystems (i.e. free-system and control-system), the sliding mode dynamics, the boundary equilibrium bifurcations, as well as the global behaviors. We prove that both subsystems may undergo backward bifurcations and the sliding domain exists. Meanwhile, it is possible that the pseudo-equilibrium exists and is globally stable, or the pseudo-equilibrium, the disease-free equilibrium and the real equilibrium are tri-stable, or the pseudo-equilibrium and the real equilibrium are bi-stable, or the pseudo-equilibrium and disease-free equilibrium are bi-stable, which depends on the threshold value and other parameter values. The global stability of the pseudo-equilibrium reveals that we may maintain the number of infected hosts at a previously given value. Moreover, the bi-stability and tri-stability indicate that whether the number of infected individuals tends to zero or a previously given value or other positive values depends on the parameter values and the initial states of the system. These results highlight the challenges in the control of environmentally-driven infectious disease.

We establish a theorem on bifurcation of limit cycles from a focus boundary equilibrium of an impacting system, which is universally applicable to prove the bifurcation of limit cycles from focus boundary equilibria in other types of piecewise-smooth systems, such as Filippov systems and sweeping processes. Specifically, we assume that one of the subsystems of the piecewise-smooth system under consideration admits a focus equilibrium that lie on the switching manifold at the bifurcation value of the parameter. In each of the three cases, we derive a linearized system which is capable of concluding the occurrence of a finite-time stable limit cycle from the above-mentioned focus equilibrium when the parameter crosses the bifurcation value. Examples illustrate how conditions of our theorems lead to closed-form formulas for the coefficients of the linearized system.

The main purpose of the paper is to investigate the effect of multiple scales in frequency domain on the complicated oscillations of Filippov system with discontinuous right-hand side. A relatively simple model based on the Chua’s circuit with periodic excitation is introduced as an example. When the exciting frequency is far less than the natural frequency, implying that an order gap between the exciting frequency and the natural frequency exists, the whole exciting term can be considered as a slow-varying parameter, based on which the bifurcations of the two subsystems in different regions divided by the nonsmooth boundary are presented. Two typical cases are considered, which correspond to different distributions of equilibrium branches as well as the related bifurcations. In the first case, periodic symmetric Hopf/Hopf-fold-sliding bursting oscillations can be obtained, in which Hopf bifurcations may cause the alternations between the quiescent states and the spiking states, while fold bifurcations connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. While the second case is the periodic symmetric fold/fold-fold-sliding bursting, where the fold bifurcations not only lead to the alternations between the quiescent states and the spiking states, but also connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. It is pointed out that, different from the bursting oscillations in smooth dynamical systems in which the bifurcations may cause the alternations between quiescent states and spiking states, in the nonsmooth system, bifurcations may not only lead to the alternations, but also connect different forms of quiescent states. Furthermore, in the Filippov system, sliding movement along the nonsmooth boundary can be observed, the mechanism of which is presented based on the analysis of the two subsystems in different regions.

Mathematical models can assist to design and understand control strategies for limited resources in Integrated Pest Management (IPM). This paper studies the dynamical behavior of a Filippov predator–prey model with periodic forcing. Firstly, bifurcation analyses are carried out to show that the Filippov predator–prey ecosystem may have very complex dynamics, i.e. the system may have periodic, quasi-periodic, chaotic solutions, as well as period doubling bifurcations. Meanwhile, the model is analyzed theoretically and numerically to understand how resource limitation and periodic forcing affect pest population outbreaks, the intersection between the initial densities (pest and natural enemy populations) and pest control has been discussed. Furthermore, the sliding surface, sliding mode dynamics, the existence and stability of sliding periodic solution of the proposed model and its application in IPM strategy are investigated. Our results show that several hidden factors can adversely affect our control strategy in limited resource and fluctuating environment. Thus, choosing a proper threshold value ET may play a decisive role in pest control, which confirms that IPM is the optimal control strategy.

Chen [2016a, 2016b] studied global dynamics of the Filippov systems $ẍ+(\alpha +\beta x^2)ẋ+x\pm sgn(x)=0$, respectively. To study the global dynamics of $ẍ+(\alpha +\beta x^2)ẋ\pm x\pm sgn(x)=0$ completely, since the dynamics of $ẍ+(\alpha +\beta x^2)ẋ-x-sgn(x)=0$ is very simple, we are only interested in the global dynamics of $ẍ+(\alpha +\beta x^2)ẋ-x+sgn(x)=0$ in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system $ẍ+(\alpha +\beta x^2)ẋ-x+\mathrm{\text{sgn}}(x)=0$ is totally different from systems $ẍ+(\alpha +\beta x^2)ẋ+x\pm \mathrm{\text{sgn}}(x)=0$.

In this paper, we report the occurrence of sliding bifurcations admitted by the memristive Murali–Lakshmanan–Chua circuit [Ishaq & Lakshmanan, 2013] and the memristive driven Chua oscillator [Ishaq et al., 2011]. Both of these circuits have a flux-controlled active memristor designed by the authors in 2011, as their nonlinear element. The three-segment piecewise-linear characteristic of this memristor bestows on the circuits two discontinuity boundaries, dividing their phase spaces into three subregions. For proper choice of parameters, these circuits take on a degree of smoothness equal to one at each of their two discontinuities, thereby causing them to behave as Filippov systems. Sliding bifurcations, which are characteristic of Filippov systems, arise when the periodic orbits in each of the subregions, interact with the discontinuity boundaries, giving rise to many interesting dynamical phenomena. The numerical simulations are carried out after incorporating proper zero time discontinuity mapping (ZDM) corrections. These are found to agree well with the experimental observations which we report here appropriately.

In order to avoid high extinction risks of prey and keep the stability of the three-species food chain model, we introduce a Filippov food chain model (FFCM) with Holling type II under threshold policy control. The threshold policy is designed to play a pivotal strategy for controlling the three species in the FFCM. With this strategy, no control is applied if the density of the prey population is less than the threshold, thus the exploitation is forbidden. However, the exploitation is permitted if the density of the prey population increases and exceeds the threshold. The dynamic behaviors and the bifurcation sets of this model including the existence and stability of different types of equilibria are discussed analytically and numerically. Moreover, the regions of sliding and crossing segments are analyzed. The dynamic behaviors of sliding mode including the bifurcation sets of pseudo-equilibria are investigated. Numerically, the bifurcation diagram and maximum Lyapunov exponents are computed and plotted to show the complex dynamics of FFCM, for instance, it has stable periodic, double periodic and chaotic solutions as well as double periodic sliding bifurcation. It is demonstrated that the threshold policy control can be easily implemented and used for stabilizing the chaotic behavior of FFCM.

We propose a nonsmooth Filippov refuge ecosystem with a piecewise saturating response function and analyze its dynamics. We first investigate some key elements to our model which include the sliding segment, the sliding mode dynamics and the existence of equilibria which are classified into regular/virtual equilibrium, pseudo-equilibrium, boundary equilibrium and tangent point. In particular, we consider how the existence of the regular equilibrium and the pseudo-equilibrium are related. Then we study the stability of the standard periodic solution (limit cycle), the sliding periodic solutions (grazing or touching cycle) and the dynamics of the pseudo equilibrium, using quantitative analysis techniques related to nonsmooth Filippov systems. Furthermore, as the threshold value is varied, the model exhibits several complex bifurcations which are classified into equilibria, sliding mode, local sliding (boundary node and focus) and global bifurcations (grazing or touching). In conclusion, we discuss the importance of the refuge strategy in a biological setting.

Considering the effectiveness of introducing the change rate of viral loads into the threshold setting policy for triggering interventions, we propose an immune-virus Filippov system with a nonlinear threshold. By developing new analytical and numerical methods, we systematically studied the rich dynamical behaviors and bifurcations of the proposed system, including the existence of three sliding segments and three pseudo-equilibria, boundary-center bifurcation, boundary-saddle bifurcation, pseudo-saddle-node bifurcation and tangency bifurcation. We further showed that the proposed system can exhibit virous structures in the coexistence of multiple steady states. Phenomena include bistability of two pseudo-equilibria, tristability and multiplestability of two pseudo-equilibria with regular equilibria or touching cycles. The modeling methods, as well as the analytical and numerical methods, can be widely applied to many other fields.

Our starting point is a discontinuous piecewise linear dynamical system in ${ℝ}^3$ with two zones that present a single invisible two-fold straight line and invariant cylinders. We broke the dynamics of this vector field to obtain a new four-parameter piecewise linear vector field with a $T$-singularity. For this vector field, we prove that the upper bound of simple crossing limit cycles (simple CLCs) is three, and provide conditions for the existence of 1, 2, or 3 simple CLCs. We also show the occurrence of three distinct bifurcations involving such simple CLCs that are derived from a bifurcation set with two parameters: (i) a Teixeira Singularity bifurcation (TS-bifurcation); (ii) a Fold bifurcation; and (iii) a Cusp bifurcation.

Dispersal of organisms between patches is a common phenomenon in ecology and plays an important role in predator–prey population dynamics. We propose a nonsmooth Filippov predator–prey model in a two-patch environment characterized by a generalist predator-driven intermittent refuge protection of an apprehensive prey along with a balanced dispersal of the prey between refuge and nonrefuge areas. By employing qualitative techniques of nonsmooth dynamical systems, we see that the switching surface is a repeller whenever the interior equilibria are virtual, causing long-term population fluctuations. We find that the level of prey vigilance and the rate of prey dispersal play pivotal roles in the total harvesting yield. We observe that a sustainable high harvesting yield is possible when the prey is less vigilant and obtain the harvesting efforts for maximum sustainable total yield (MSTY). We further modify the model by considering a continuous threshold predator-driven prey dispersal and show that the model exhibits a Hopf bifurcation when the level of prey vigilance exceeds some critical threshold value. By comparing the dynamics of the two models we see that for a sustainable high harvesting yield of the system with continuous threshold dispersal, the prey needs to be highly vigilant compared to that of the system with intermittent dispersal of the prey. Further, we find numerically that the estimated MSTY from both models remains the same.

In this paper, we propose a Filippov switching model which is composed of the Lorenz and Chen systems. By employing the qualitative analysis techniques of nonsmooth dynamical systems, we show that the new Filippov system not only inherits the properties of the Lorenz and Chen systems but also presents new dynamics including new chaotic attractors such as four-wing butterfly attractor, Lorenz attractor with sliding segments, etc. In particular, we find that different new attractors can coexist such as the coexistence of two-point attractors and chaotic attractor, the coexistence of two-point attractors and quasi-periodic solution, the coexistence of transient transition chaos and quasi-periodic solution. Furthermore, nonsmooth bifurcations and numerical analyses reveal that the proposed Filippov system has a series of new sliding bifurcations including a symmetric pair of sliding mode bifurcations, a symmetric pair of sliding Hopf bifurcations, and a symmetric pair of Hopf-like boundary equilibrium bifurcations.

Filippov systems have found applications in various fields. This paper mainly studies five Lotka–Volterra models of Filippov type, including a competitive system with linear interaction between two species, a competitive system with Holling type II or type III functional response and a symbiosis system with Holling type II or type III functional response. We investigate the stability of all equilibria and the boundary equilibrium bifurcations of these systems, either a persistence bifurcation or a nonsmooth fold bifurcation. We present the numerical simulation results for each case. Consequently, based on both theory and simulation, we analyze the ecological aspects under the intervention of harvesting at different prescribed thresholds.

In this article, we consider a SIV infectious disease control system with two-threshold guidance, in which infection rate and vaccination rate are represented by a piecewise threshold function. We analyze the global dynamics of the discontinuous system using the theory of differential equations with discontinuous right-hand sides. We find some dynamical behaviors, including the disease-free equilibrium and endemic equilibria of three subsystems, a globally asymptotically stable pseudo-equilibrium and two endemic equilibria bistable, one of the two pseudo-equilibria or pseudo-attractor is stable. Conclusions can be used to guide the selection of the most appropriate threshold and parameters to achieve the best control effect under different conditions. We hope to minimize the scale of the infection so that the system can eventually stabilize at the disease-free equilibrium, pseudo-equilibrium or pseudo-attractor, corresponding to the disease disappearing or becoming endemic to a minimum extent, respectively.