Boundary Equilibrium Bifurcations Research Articles

Nonsmooth folds as tipping points

A nonsmooth fold occurs when an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same type. We show that beyond the bifurcation, the leading-order truncation to the system, in general, has no bounded invariant set. This is proved for boundary equilibrium bifurcations of Filippov systems, hybrid systems, and continuous piecewise-smooth ordinary differential equations, and grazing-type events for which the truncated form is a continuous piecewise-linear map. The omitted higher-order terms are expected to be incapable of altering the local dynamics qualitatively, implying the system has no local invariant set on one side of a nonsmooth fold, and we demonstrate this with an example. Thus, if the equilibrium or limit cycle is attracting, the bifurcation causes the local attractor of the system to tip to a new state. The results also help explain global aspects of bifurcation structures of the truncated systems.

Stability and bifurcations for a 3D Filippov SEIS model with limited medical resources

In this paper, a three-dimensional (3D) Filippov SEIS epidemic model is proposed to characterize the impact of limited medical resources on disease transmission with discontinuous treatment functions. Qualitative analysis of non-smooth dynamical behaviors are performed on two subsystems and sliding modes. Criteria on the stability of various kinds of feasible equilibria and bifurcations, e.g., saddle-node bifurcation, transcritical bifurcation, and boundary equilibrium bifurcation, are established. The theoretical results are illustrated by numerical simulation, from which we find there could exist bistable phenomena, e.g., endemic and pseudo-equilibria, endemic equilibria of the two subsystems, or endemic and disease-free equilibria, even the basic reproduction numbers of two subsystems are less than 1. The disease spread is dependent both on the limited medical resources and latent compartment, which are more beneficial to effective disease control than planar Filippov and smooth models.

Frequency switching leads to distinctive fast–slow behaviors in Duffing system

Frequency switching leads to distinctive fast–slow behaviors in Duffing system

Mosquito suppression via Filippov incompatible insect technique

Mosquito suppression via Filippov incompatible insect technique

Two-Parameter Bifurcations and Hidden Attractors in a Class of 3D Linear Filippov Systems

We take into consideration two different kinds of two-parameter bifurcations in a class of 3D linear Filippov systems, namely pseudo-Bautin bifurcation and boundary equilibrium bifurcations for two scenarios. The bifurcation conditions for generating rich dynamic behaviors are established. The main objective is to investigate the effects of two parameters interacting simultaneously on a variety of dynamic phenomena. In order to analyze the pseudo-Bautin bifurcation, we build the Poincaré map and analyze the number of fixed points whose types are related to the crossing limit cycles. In order to analyze boundary equilibrium bifurcations for two scenarios, we perform an analysis on the existence and admissibility of equilibria. Besides, a comprehensive investigation on hidden attractors induced by boundary equilibrium bifurcations is conducted. The novelty resides in overcoming the constraints of previous studies that solely take into account the dynamics of individual parameter variations. We innovatively characterize the two-parameter bifurcation mechanism of a new class of Filippov systems, and qualitatively demonstrate the coexistence of hidden attractor and stable pseudo-equilibrium.

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.

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>

Threshold control strategy for a Filippov model with group defense of pests and a constant-rate release of natural enemies.

In this paper, we establish an integrated pest management Filippov model with group defense of pests and a constant rate release of natural enemies. First, the dynamics of the subsystems in the Filippov system are analyzed. Second, the dynamics of the sliding mode system and the types of equilibria of the Filippov system are discussed. Then the complex dynamics of the Filippov system are investigated by using numerical analysis when there is a globally asymptotically stable limit cycle and a globally asymptotically stable equilibrium in two subsystems, respectively. Furthermore, we analyze the existence region of a sliding mode and pseudo equilibrium, as well as the complex dynamics of the Filippov system, such as boundary equilibrium bifurcation, the grazing bifurcation, the buckling bifurcation and the crossing bifurcation. These complex sliding bifurcations reveal that the selection of key parameters can control the population density no more than the economic threshold, so as to prevent the outbreak of pests.

Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response.

In this paper, we consider the dynamics of a slow-fast Bazykin's model with piecewise-smooth Holling type Ⅰ functional response. We show that the model has Saddle-node bifurcation and Boundary equilibrium bifurcation. Furthermore, it is also proven that the model has a homoclinic cycle, a heteroclinic cycle or two relaxation oscillation cycles for different parameters conditions. These results imply the dynamical behavior of the model is sensitive to the predator competition rate and the initial densities of prey and predators. In order to support the theoretical analysis, we present some phase portraits corresponding to different values of parameters by numerical simulation. These phase portraits include two relaxation oscillation cycles, an unstable relaxation oscillation cycle surrounded by a stable homoclinic cycle; the coexistence of a heteroclinic cycle and an unstable relaxation oscillation cycle. These results reveal far richer and much more complex dynamics compared to the model without different time scale or with smooth Holling type Ⅰ functional response.

Dynamical analysis in a piecewise smooth predator–prey model with predator harvesting

The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator–prey model with predator harvesting. We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). It is shown that boundary equilibrium bifurcation and sliding–grazing bifurcation can happen as the threshold value varies. We provide analytical analysis to prove the existence of sliding limit cycles and sliding homoclinic cycles, the coexistence of them with standard limit cycles. Some numerical simulations are given to demonstrate our results.

Global studies on a continuous planar piecewise linear differential system with three zones

This paper is concerned with the global dynamics of a continuous planar piecewise linear differential system with three zones, where the dynamic of the one of the exterior linear zones is saddle and the remaining one is anti-saddle. We give all global phase portraits in the Poincaré disc and the complete bifurcation diagram including boundary equilibrium bifurcation curves, degenerate boundary equilibrium bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curves. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results.

Dynamics and Bifurcations in Filippov Type of Competitive and Symbiosis Systems

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.

Complex Dynamics and Sliding Bifurcations of the Filippov Lorenz–Chen System

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.

Dynamics of a non-smooth model of prostate cancer with intermittent androgen deprivation therapy

Intermittent androgen deprivation therapy (IADT) is widely used in the clinic treatment for prostate cancer. We propose a Filippov system in this paper, which defines the following threshold policy: the androgen deprivation therapy (ADT) is implemented once the concentration of androgen-dependent (AD) cells exceeds the threshold level E T and it is suspended otherwise. To better understand the effect of the IADT on the evolution of prostate cancer, we perform a comprehensive qualitative analysis of global dynamics of the targeted model. As the threshold value varies, our model admits multistability of three equilibria, bistability of two equilibria, stability of the regular equilibrium or the pseudo-equilibrium, and the boundary equilibrium bifurcation. The steady-state regimes include high concentration, low concentration and no androgen-independent cells. This highlights the critical role of threshold values and the initial condition of prostate cancer cells of patients. The main results indicate that the concentration of AD cells and AI cells can be contained at a relatively satisfactory level, which can help to make treatment schedule and improve treatment outcomes for patients with prostate cancer. • A non-smooth model was proposed to explore the effect of intermittent therapy. • Multistability of three or two equilibria occurs for the targeted model. • The concentration of cancer cells can be eradicated or contained at a high/low level. • A proper threshold policy can assist in improving the treatment outcome for patients.

Dimension reduction for slow-fast, piecewise-linear ODEs and obstacles to a general theory

This paper concerns systems of ODEs that are continuous, piecewise-linear (with two pieces) and slow-fast. The limiting slow dynamics occurs on a piecewise-linear critical manifold that corresponds to equilibria of the fast dynamics (layer equations). When the critical manifold is attracting for each piece of the layer equations, one would like to know when the full dynamics of the system near this manifold is a regular perturbation of the limiting slow dynamics (reduced system). We identify three phenomena unique to the piecewise-linear setting that obstruct such an extension of Fenichel theory: at the boundary (switching manifold) the equilibrium of the layer equations can be unstable, coincide with another attractor, or attract orbits in a chaotic fashion. The first two phenomena (at least) can inhibit a regular perturbation and lead to the creation of canards. If a regular perturbation does occur for the piecewise-linear approximation to a boundary equilibrium bifurcation, dimension reduction can be achieved and we illustrate this with a three-dimensional model of ocean circulation. We also introduce a slow-fast version of the observer canonical form for piecewise-linear ODEs that simplifies the analysis. • General aspects of continuous, piecewise-linear, slow-fast, ordinary differential equations are studied. • Three phenomena are identified that may prevent the dynamics being a regular perturbation of the reduced dynamics. • A slow-fast version of the observer canonical form is introduced. • The results are illustrated with an ocean circulation model.

The discontinuous limit case of an archetypal oscillator with a constant excitation and van der Pol damping

We investigate global dynamics of the discontinuous limit case of an archetypal oscillator with a constant excitation that is a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. The discontinuous dynamical system can exhibit three limit cycles and rich nonlinear phenomena, including boundary equilibrium bifurcation, Hopf bifurcation, grazing bifurcation, pseudo saddle–node loop bifurcation and double crossing limit cycle bifurcation. Because of destroyed symmetry for this system, it exhibits more complex and abundant dynamics than the smooth oscillator with van der Pol damping and can appear new dynamical phenomena, such as coexistence of a crossing limit cycle and a pseudo homoclinic loop, coexistence of two crossing limit cycles and the existence of pseudo saddle–node loops. The bifurcation diagram of the discontinuous system is shown in parameter space and all global phase portraits in the Poincaré disk are sketched in the phase plane. • Complete global phase portraits in the Poincare disk and give the bifurcation diagram. • Some new criterions on limit cycles of a discontinuous dynamical system are given. • Rich nonlinear phenomena can occur.

Bifurcation dynamics on the sliding vector field of a Filippov ecological system

A Filippov crop-pest-natural enemy ecological system with threshold switching surface related to the pest control is developed, which has been completely analyzed by employing the qualitative techniques of non-smooth dynamical systems. The main results reveal that the proposed switching model can have multiple pseudo-equilibria in the sliding region, which result in rich bifurcations in the sliding region including saddle-node, Hopf, Bogdanov-Takens and Hopf-like boundary equilibrium bifurcations. Moreover, the pseudo-periodic solution (or pseudo-homoclinic loop) can be generated in the sliding region through a Hopf bifurcation (or a homoclinic bifurcation), which can collide with the tangential lines at the cusp singularities and finally disappears as parameter varies. This reveals that although the system can stabilize on the sliding region to achieve the purpose of pest control, there are complex dynamical behaviors and sliding bifurcations within the sliding region. Furthermore, as the threshold level varies, the model exhibits the interesting global sliding bifurcations including grazing bifurcation, buckling bifurcation, crossing bifurcation, homoclinic bifurcation to a pseudo-saddle, period-halving bifurcation and chaotic dynamics. 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.

Global dynamics for a Filippov system with media effects.

In the process of spreading infectious diseases, the media accelerates the dissemination of information, and people have a deeper understanding of the disease, which will significantly change their behavior and reduce the disease transmission; it is very beneficial for people to prevent and control diseases effectively. We propose a Filippov epidemic model with nonlinear incidence to describe media's influence in the epidemic transmission process. Our proposed model extends existing models by introducing a threshold strategy to describe the effects of media coverage once the number of infected individuals exceeds a threshold. Meanwhile, we perform the stability of the equilibriua, boundary equilibrium bifurcation, and global dynamics. The system shows complex dynamical behaviors and eventually stabilizes at the equilibrium points of the subsystem or pseudo equilibrium. In addition, numerical simulation results show that choosing appropriate thresholds and control intensity can stop infectious disease outbreaks, and media coverage can reduce the burden of disease outbreaks and shorten the duration of disease eruptions.

Bursting Oscillations as well as the Mechanism in a Filippov System with Parametric and External Excitations

The main purpose of this paper is to explore the bursting oscillations as well as the mechanism of a parametric and external excitation Filippov type system (PEEFS), in which different types of bursting oscillations such as fold/nonsmooth fold (NSF)/fold/NSF, fold/NSF/fold and fold/fold bursting oscillations can be observed. By employing the overlap of the transformed phase portrait and the equilibrium branches of the generalized autonomous system, the mechanisms of the bursting oscillations are investigated. Our results show that the fold bifurcation and the boundary equilibrium bifurcation (BEB) can cause the transitions between the quiescent states and repetitive spiking states. The oscillating frequencies of the spiking states can be approximated theoretically by their occurring mechanisms, which agree well with the numerical simulations. Furthermore, some nonsmooth evolutions are investigated by employing differential inclusions theory, which reveals that the positional relationship between the points of the trajectory interacting with the nonsmooth boundary and the related sliding boundary of the nonsmooth system may affect the nonsmooth evolutions.

Global dynamics for a Filippov epidemic system with imperfect vaccination

Given imperfect vaccination we extend the existing non-smooth models by considering susceptible and vaccinated individuals enhance the protection and control strategies once the number of infected individuals exceeds a certain level. On the basis of global dynamics of two subsystems, for the formulated Filippov system, we examine the sliding mode dynamics, the boundary equilibrium bifurcations, and the global dynamics. Our main results show that 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 depend on the threshold value and other parameter values. The global stability of the disease-free equilibrium or pseudo-equilibrium reveals that we may eradicate the disease or maintain the number of infected individuals at a previously given value. Further, the bi-stability and tri-stability imply 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. This emphasizes the importance of threshold policy and challenges in the control of infectious diseases if without perfect vaccines.