Filippov System Research Articles

Rich dynamics of a reaction–diffusion Filippov Leslie–Gower predator–prey model with time delay and discontinuous harvesting

To reflect the harvesting effect, a nonsmooth Filippov Leslie–Gower predator–prey model is proposed. Unlike traditional Filippov models, the time delay and reaction–diffusion under the condition of homogeneous Neumann boundary are considered in our system. The stability of equilibrium and the existence of the spatial Hopf bifurcation of the subsystems at the positive equilibrium are investigated. Furthermore, a comprehensive analysis is conducted on the sliding mode dynamics as well as the regular, virtual, and pseudoequilibria. The findings reveal that our Filippov system exhibits either a globally asymptotically stable regular equilibrium, a globally asymptotically stable time periodic solution, or a globally asymptotically stable pseudoequilibrium, contingent upon the specific values of the time delay and threshold level. A boundary point bifurcation, which transform a stable equilibrium point or periodic solution into a stable pseudoequilibrium, is demonstrated to emphasize the impact of time delay on our Filippov system and the significance of threshold control. Meanwhile, two kinds of global sliding bifurcations are exhibited, which sequentially transform a stable periodic solutions below the threshold into a grazing, sliding switching, and crossing bifurcations, depending on changes in the time delay or threshold level. Our results indicate that bucking bifurcation and crossing bifurcation pose significant challenges to the control of our Filippov system.

Discontinuous harvesting policy in a Filippov system involving prey refuge

Discontinuous harvesting policy in a Filippov system involving prey refuge

Nonsmooth dynamics of a Filippov predator–prey ecological model with antipredator behavior

This article proposes a class of nonsmooth Filippov pest–predator ecosystems with intermittent control strategies based on the pest’s antipredator behavior. aiming to investigate the influence of control strategies and switching thresholds on pest control. First, a comprehensive theoretical analysis of various equilibria within the Filippov system is undertaken, emphasizing the presence and stability of sliding mode dynamics and pseudoequilibrium. Secondly, through numerical simulations, the article discusses boundary-focus, boundary-node, and boundary-saddle bifurcation. Finally, the nonexistence of limit cycles in the Filippov system is theoretically studied. The research indicates that the solution trajectories of the model ultimately stabilize either at the real equilibria or at pseudoequilibrium on the model’s switching surface. Moreover, when the model has multiple coexisting real equilibrium and pseudoequilibrium, the pest-control strategy is correlated with the initial density of both the pest and the predator population.

Sliding dynamics of a Filippov ecological system with nonlinear threshold control and pest resistance

By applying the integrated pest management strategy to address pest control problems, Filippov pest-natural enemy systems have been extensively studied. However, little attention has been paid to pest changing rate and pest resistance in these studies. Therefore, this paper proposes a complex and more realistic Filippov pest-natural enemy system with curved discontinuity boundary by considering Beddington–DeAngelis functional response and pest resistance, as well as incorporating pest changing rate in the threshold control index. Based on the relevant theory of the Filippov system, we analyze a sextic equation and obtain that the proposed Filippov system can have at most six sliding segments. By using qualitative techniques, we discuss the sliding segment and pseudo-equilibrium bifurcations. Numerically, we investigate the local and global sliding bifurcations. It can be detected that one new local sliding bifurcation occurs, which could be called the pseudo-Hopf bifurcation but differs from the traditional pseudo-Hopf bifurcation as it involves two sliding segments with opposite stabilities and generates a sliding limit cycle. We also discover two interesting global sliding bifurcations with novel bifurcation processes, which may be termed triple limit cycle bifurcations as they both relate to the appearance of three nested limit cycles.

Non-smooth dynamics of a fishery model with a two-threshold harvesting policy

The non-linear dynamical systems theory helps implement regulatory measures to control the growth and evolution of various populations. While invasion by alien fish species is an emerging threat to native fish species in marine ecosystems, a suitable fishery management protocol needs to be incorporated in marine protected areas (MPAs) to mitigate the problem. We propose a policy of selective harvesting whenever the density of a species crosses a predefined threshold. An ecosystem with such a harvesting policy is modelled by a piece-wise smooth prey-predator type fishery model, where a control function defined by two thresholds evokes signals to cease or commence harvesting. This work reports the dynamics and bifurcations of a Filippov system, including the possibility of sliding mode dynamics. We show that the variations in the growth rate and the harvesting rate of the invasive fish species can lead to hysteresis through saddle–node and transcritical bifurcations and the appearance of a stable homoclinic orbit. We obtain the conditions for sliding domains, regular or virtual equilibria, boundary equilibria, pseudo-equilibria, and tangent points. Depending on the parameter choice, the system can stabilize at a regular equilibrium, a pseudo-equilibrium, or a pseudo-attractor, exhibiting multistability. We report the occurrence of regular or virtual equilibrium bifurcation, boundary node bifurcation and pseudo-saddle–node bifurcation. Our findings indicate that proper harvesting strategies can prevent the proliferation of invasive fish species in MPAs.

Regularization of the Boundary Equilibrium Bifurcation in Filippov System with Rich Discontinuity Boundaries

This paper studies a particular type of planar Filippov system that consists of two discontinuity boundaries separating the phase plane into three disjoint regions with different dynamics. This type of system has wide applications in various subjects. As an illustration, a plant disease model and an avian-only model are presented, and their bifurcation scenarios are investigated. By means of the regularization approach, the blowing up method, and the singular perturbation theory, we provide a different way to analyze the dynamics of this type of Filippov system. In particular, the boundary equilibrium bifurcations of such systems are studied. As a consequence, the nonsmooth fold bifurcation becomes a saddle-node bifurcation, while the persistence bifurcation disappears after regularization.

FILIPPOV TYPE PREY–PREDATOR SYSTEM FOR SELECTIVE HARVESTING OF PREY

In this paper, the dynamics of a Filippov system with logistically growing prey and Holling Type II predation is explored. The selective nonlinear harvesting of prey is carried out when the ratio of prey to predator is above a specified threshold value. In order to prevent the extinction of both the species and sustainable economic gains, no harvesting is allowed when the ratio is below a threshold value. The equilibrium states and their stability for the individual smooth are analyzed. Filippov convex method is applied to determine the dynamics on the boundary. The sliding mode dynamics is scrutinized, and it is observed that there is no escaping region. The existence of boundary points, tangent points and pseudo-equilibrium points are established. The conditions are obtained for the visibility of the tangent points. The complex behavior within the sliding region is studied by experimenting with different initial conditions. The increase in harvesting effort may lead to extinction of both the species through the sliding region of the boundary. The change in the stability behavior of both the interior equilibrium states is investigated with respect to model parameters. Border cross bifurcates with threshold value as the bifurcation parameter is shown for both the interior equilibrium states.

Piecewise immunosuppressive infection model with viral logistic growth and effector cell-guided therapy

<abstract><p>This work investigated a piecewise immunosuppressive infection model that assessed the effectiveness of implementing this therapeutic regimen once the effector cell count falls below a specific threshold level by introducing a threshold strategy. The sliding mode dynamics, global dynamics, and boundary equilibrium bifurcations of the Filippov system were examined based on the global dynamics of the two subsystems. Our primary findings indicate that the HIV viral loads and effector cell counts can be stabilized within the required predetermined level. This outcome depends on the threshold level, immune intensity, and the initial values of the system. Therefore, properly combining these key factors makes it possible to effectively curb the abnormal increase of virus and keep the effector cells at a reliable level. This approach maximizes the controllable range of the HIV. The proposed switching system incorporating pseudo-equilibrium exhibits three types of equilibriums that could be bistable or tristable. It means there is a possibility of controlling the virus after administering therapy if the immune intensity $ c $ is limited within the range of the post-treatment control threshold and the elite control threshold when $ {R_0} > {R_{{c_1}}} > {R_{{c_2}}} > 1 $.</p></abstract>

New results on finite-time projective synchronization for memristor-based hybrid delayed BAM neural networks with applications to DNA image encryption

<abstract> <p>With the popularization of digital image technology, image information has inevitably developed to involved the disclosure of personal privacy; in this study, a color image encryption algorithm was designed to encrypt and decrypt images by using chaotic sequences of a class of memristor-based hybrid delayed bidirectional associative memory neural networks (MHDBAMNNs) to protect images from illegal acquisition and use. Additionally, the discontinuity problem of the right-hand side of the Filippov system due to the hopping property of the memristor has been treated by using differential inclusion and set-valued mapping theories, and a sufficient criterion for guaranteeing the synchronization of finite-time projections derived based on the drive-response concept, Lyppunov stability theorem, and inequality technique. To improve the security performance, a color image encryption algorithm based on a combination of Chen's hyperchaotic system and a DNA codec operation was adopted, also, the robustness and validity of our proposed approach was demonstrated through image performance analysis. Furthermore, the potential application of the model in secure transmission has been explored.</p> </abstract>

Qualitative analysis of a Filippov wild-sterile mosquito population model with immigration.

Effectively combating mosquito-borne diseases necessitates innovative strategies beyond traditional methods like insecticide spraying and bed nets. Among these strategies, the sterile insect technique (SIT) emerges as a promising approach. Previous studies have utilized ordinary differential equations to simulate the release of sterile mosquitoes, aiming to reduce or eradicate wild mosquito populations. However, these models assume immediate release, leading to escalated costs. Inspired by this, we propose a non-smooth Filippov model that examines the interaction between wild and sterile mosquitoes. In our model, the release of sterile mosquitoes occurs when the population density of wild mosquitoes surpasses a specified threshold. We incorporate a density-dependent birth rate for wild mosquitoes and consider the impact of immigration. This paper unveils the complex dynamics exhibited by the proposed model, encompassing local sliding bifurcation and the presence of bistability, which entails the coexistence of regular equilibria and pseudo-equilibria, as crucial model parameters, including the threshold value, are varied. Moreover, the system exhibits hysteresis phenomena when manipulating the rate of sterile mosquito release. The existence of three types of limit cycles in the Filippov system is ruled out. Our main findings indicate that reducing the threshold value to an appropriate level can enhance the effectiveness of controlling wild insects. This highlights the economic benefits of employing SIT with a threshold policy control to impede the spread of disease-carrying insects while bolstering economic outcomes.

Periodic trajectories in planar discontinuous piecewise linear systems with only centers and with a nonregular switching line

In this paper periodic trajectories of dynamical systems presenting discontinuities are studied. The considered model consists of two distinct linear differential systems, each one containing a single equilibrium point of centre type. Each system is defined on disjoint regions of the plane, the separation line is a union of two half-straight lines contained on the coordinate axes. The obtained differential system is non-smooth, so we apply Filippov’s theory to study the transitions from one dynamical system to another. The combination of the two linear plus the Filippov system acting on the separation line generates a nonlinear regime observed by the presence of limit cycles, sliding and tangential periodic trajectories as well as the coexistence of such objects. In theorem we establish the location, stability and hyperbolicity of limit cycles for certain classes of the considered model. In theorem we perform the global analysis of a representative model through bifurcation theory to analyse the birth of limit cycles, sliding periodic trajectories and tangential ones. We also provide some results addressing the coexistence of periodic trajectories and two potential physical interpretations of the model considered in the paper, one addressing nonlinear oscillations and the other considering slow-fast systems of neuron models. The main techniques employed to obtain the results are first integrals, Poincaré half return maps, and elements of bifurcation theory.

The time-freezing reformulation for numerical optimal control of complementarity Lagrangian systems with state jumps

This paper introduces a novel time-freezing reformulation and numerical methods for optimal control of complementarity Lagrangian systems (CLS) with state jumps. We cover the difficult case when the system evolves on the boundary of the dynamic’s feasible set after the state jump. In nonsmooth mechanics, this corresponds to inelastic impacts. The main idea of the time-freezing reformulation is to introduce a clock state and an auxiliary dynamical system whose trajectory endpoints satisfy the state jump law. When the auxiliary system is active, the clock state is not evolving, hence by taking only the parts of the trajectory when the clock state was active, we can recover the original solution. The resulting time-freezing system is a Filippov system that has jump discontinuities only in the first time derivative instead of the trajectory itself. This enables one to use the recently proposed Finite Elements with Switch Detection (Nurkanović et al., 2022), which makes high accuracy numerical optimal control of CLS with impacts and friction possible. We detail how to recover the solution of the original system and show how to select appropriate auxiliary dynamics. The theoretical findings are illustrated on a nontrivial numerical optimal control example of a hopping one-legged robot.

New Inequality Approaches for Fixed-Time Stability Lemmas and Application to Discontinuous CGNNs With Nondifferentiable Delays

This article proposes fixed-time stability lemmas for the Filippov system via some new inequality approaches. The adopted method no longer needs to integrate the Lyapunov function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$</tex-math> </inline-formula> on the two integral intervals, which is quite different from the existing ones. Some new estimations of the settling times are provided. Also, the steepness exponents of the Lyapunov function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$</tex-math> </inline-formula> in the previous fixed-time stability lemmas are improved. In order to further study the nondifferentiable delayed neural networks modeled by the Filippov system, a class of discontinuous uncertain Cohen–Grossberg neural networks (CGNNs) with mixed delays is formulated and the distributed delays are nondifferentiable, which is more general. Due to the existence of nondifferentiable distributed delays, the existence of the periodic solutions is proved by Kakutani’s fixed-point theorem before considering the stability. By virtue of the obtained fixed-time stability lemmas and the constructed delay-product-type Lyapunov–Krasovskii functional, the fixed-time stabilization is obtained via a no-chattering controller. Clearly, the designed controller does not contain integral terms and delay terms for dealing with the time delays in the closed-loop system, which is more simplified and practical. Finally, two examples help examine the correctness of the main results.

Dynamic Behaviours and Bifurcation Analysis of a Filippov Ecosystem with Fear Effect

Evidences of the fear effect of the prey are well documented, which can greatly affect the dynamics of the predator-prey system. In this study, considering that the fear effect of the prey is triggered on as the density of the predator reaches and exceeds a threshold value, we develop a Filippov system of predator-prey model with the fear effect. In addition, we also include a modify factor of the growth rate of the prey when they adopt the antipredator behaviours due to the fear effect. We initially analyze the dynamics of the two subsystems, including the existence and stability of the equilibria. Utilizing the theory of the Filippov system, we discuss the sliding dynamics, i.e., the existence of sliding region and sliding equilibria. By choosing the threshold as the bifurcation parameter, we investigate the bifurcations near the regular equilibria. The solution curve has three cases: crossing the threshold curve, sliding on the threshold curve, and approaching the pseudoequilibrium. Finally, we numerically verified the existence of the global sliding bifurcation near the regular equilibrium and also the touching bifurcation.

Complex dynamics of a non-smooth temperature-sensitive memristive Wilson neuron model

The development of neurodynamics emphasizes more accurate estimation and prediction of neuronal electrical activities in complicated physiological environments, which highlights the importance of reliable multifunctional neuronal modeling. Considering that temperature is a crucial factor in regulating the conductance and memory effect of ion channels, a non-smooth feedback strategy for temperature-dependent sodium and potassium currents is proposed. Accordingly, a Filippov-type Wilson neuron model is established to estimate the firing features of neocortical neurons under the effects of temperature and magnetic induction. The theoretical conditions for the existence and bifurcation of the equilibrium point of subsystems are clarified, further, the comb-shaped fractal structure and bistable firing modes are discovered by multiple numerical methods. The sufficient and necessary conditions of crossing, grazing, and sliding motions are presented qualitatively based on the flow switching theory. Importantly, the mechanism and evolutive rule of the self-excited and hidden sliding firing modes are revealed by employing the fast–slow analysis method. Moreover, a common approach for calculating the largest Lyapunov exponent of non-smooth systems based on Hamilton energy is designed, so that the global bifurcation patterns and multistability of the Filippov system are efficiently identified. The modeling scheme and the obtained results provide significant theoretical support for designing and optimizing intelligent systems.

Dynamics of a non-smooth pest-natural enemy model with the threshold control strategy

Pest issues have always been the focus of attention in agriculture. The Integrated Pest Management(IPM) method is currently the most popular way to be applied for pest control. In this study, according to the IPM strategy, we regard pest quantity as a threshold index and extend the Leslie-Gower model into a non-smooth Filippov system through combining chemical and biological control. To maintain the pest population at or below the given economic threshold(ET), we investigate the global dynamics of the proposed model, including the existence of sliding mode and various equilibria, sliding dynamics and bifurcations, and global stability of equilibria. The result shows that desired equilibria can be globally stable under some conditions, meaning that our control tactics work. In particular, the case where our strategy fails to be effective arouses interest. In the end, the biological implications of the results are discussed and given in detail.

Population dynamics with protection and harvesting of a species

In this work we study the dynamics associated to the interaction of juveniles and adults of the same species, where the harvesting of adults is not allowed when the number of adults is below a critical value. This study is carried out by bifurcation analysis, for a Filippov system, in relation to two parameters: harvesting and protection of the adult species.

Stability and finite/fixed-time attractivity of time-delayed filippov system: application to switched neural networks

Stability and finite/fixed-time attractivity of time-delayed filippov system: application to switched neural networks

Global dynamics of a Filippov system with general parameters and saddle structure of a regular-SN

In this paper we investigate the global dynamics of a planar 3-parametric Filippov system with saddle structure of a regular-saddle-node for general parameters. Our work generalizes previous publications in four aspects: vertical switching line to non-vertical, fixed regular direction to non-fixed, small parameters to general, local dynamics to global. Analyzing the qualitative properties of singular points on switching line and in whole Poincaré disc as well as all kinds of closed loops, we obtain the global bifurcation diagram and all corresponding global phase portraits. From our result, in the case of small parameters we find the heteroclinic bifurcation in this system, which does not appear in previous publications. Moreover, in the case of general parameters we find some new bifurcations in this system not appearing for small parameters such as double tangency bifurcation, pseudo-saddle-node bifurcation and some new bifurcation phenomena of boundary node and boundary saddle.

Occurrence of non-smooth bursting oscillations in a Filippov system with slow-varying periodic excitation

Occurrence of non-smooth bursting oscillations in a Filippov system with slow-varying periodic excitation