Filippov System Research Articles

A piecewise model of virus-immune system with effector cell-guided therapy

• We proposed a piecewise model with particular focus on the effector cellguided therapy. • The sliding domain and the sliding dynamics of the model have been investigated. • The global dynamics of the model was discussed using analytical technique of Filippov system. • Proper combination of threshold and initial values can preclude HIV virus exceptionally growing. • The controllable region can be extremely enlarged with appropriate conditions. To assess the effectiveness of structured treatment interruptions (STIs) for treating HIV patients, recent clinical studies have investigated the plasma HIV RNA and CD4 T cell-guided therapies. Here, we consider a piecewise model with a particular focus on the effector cell-guided therapy. We prove that there always exist two sliding segments, and under certain conditions two pseudo-equilibria can occur simultaneously. We also examine the global dynamics of the model by using the theory of sliding dynamics. Compared with our previous work on the system with HIV RNA-guided therapy, we show that the model for effector cell-guided therapy captures much more complex dynamics, including the coexistence of multiple attractors and infinitely many possible topological structures of the attractors. Our analysis show that with appropriate use of the effector cell-guided therapy, we can enlarge the controllable region substantially in a variety of ways for different attractors. We also note that the virus load may grow to infinity for certain initial conditions (which are patient dependent). Therefore, an optimal control strategy for the effector cell-guided therapy should be personalized, by taken into account individual patient data.

Hopf and Homoclinic bifurcations on the sliding vector field of switching systems in [formula omitted]: A case study in power electronics

In this paper, Hopf and homoclinic bifurcations that occur in the sliding vector field of switching systems in R3 are studied. In particular, a dc–dc boost converter with sliding mode control and washout filter is analyzed. This device is modeled as a three-dimensional Filippov system, characterized by the existence of sliding movement and restricted to the switching manifold. The operating point of the converter is a stable pseudo-equilibrium and it undergoes a subcritical Hopf bifurcation. Such a bifurcation occurs in the sliding vector field and creates, in this field, an unstable limit cycle. The limit cycle is connected to the switching manifold and disappears when it touches the visible–invisible two-fold point, resulting in a homoclinic loop which itself closes in this two-fold point. The study of these dynamic phenomena that can be found in different power electronic circuits controlled by sliding mode control strategies are relevant from the viewpoint of the global stability and robustness of the control design.

Hopf bifurcation for a discontinuous HTLV-1 model

Developing accurate mathematical models for host immune response in immunosuppressive diseases such as HIV and HTLV-1 are essential to achieve an optimal drug therapy regime. Since for HTLV-1 specific CTL response typically occurs after a time lag, we consider a discontinuous response function to better describe this lagged response during the early stage of the infectious, thus the system of HTLV-1 model will be a discontinuous system. For analyzing the dynamic of the system we use Filippov theory and find conditions in which the Filippov system undergoes a Hopf bifurcation. The Hopf bifurcation help us to find stable and unstable periodic oscillations and can be used to predict whether the CTL response can return to a steady state condition. Also, Hopf bifurcation in sliding mode is investigated. In this case the solutions will remain in the hyper-surface of discontinuity and as a consequence the disease cannot progress, at least for a long time. Finally we use numerical simulations to demonstrate the results by example.

Contraction analysis of switched systems via regularization

We study incremental stability and convergence of switched (bimodal) Filippov systems via contraction analysis. In particular, by using results on regularization of switched dynamical systems, we derive sufficient conditions for convergence of any two trajectories of the Filippov system between each other within some region of interest. We then apply these conditions to the study of different classes of Filippov systems including piecewise smooth (PWS) systems, piecewise affine (PWA) systems and relay feedback systems. We show that contrary to previous approaches, our conditions allow the system to be studied in metrics other than the Euclidean norm. The theoretical results are illustrated by numerical simulations on a set of representative examples that confirm their effectiveness and ease of application.

The Filippov Approach for Predator-Prey System Involving Mixed Type of Functional Responses

In this paper, a Filippov type model is proposed for the predator-prey system. The smooth subsystems of the proposed model admit regular and virtual equilibrium points and their dynamics is explored. The tangent points, boundary equilibrium and pseudo-equilibrium are obtained on discontinuity boundary. Equation of sliding motion is obtained using the Filippov’s convex method and sliding mode dynamics is discussed. It has been shown that the Filippov system admits pseudo-equilibrium, only if virtual equilibrium of two subsystems coexist. The value of threshold parameter is computed for which tangent point and boundary equilibrium collide. The two parameter bifurcation diagram is drawn to show the existence of regular and virtual equilibrium points in different regions. The existence of boundary equilibrium bifurcation is investigated through numerical simulation, as value of threshold parameter varies.

Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure

In reality, the outbreak of emerging infectious diseases including SARS, A/H1N1 and Ebola are accompanied by the common cold and flu. The selective treatment measure for mitigating and controlling the emerging infectious diseases should be implemented due to limited medical resources. However, how to determine the threshold infected cases and when to implement the selective treatment tactics are crucial for disease control. To address this, we derive a non-smooth Filippov system induced by selective treatment measure. The dynamic behaviors of two subsystems have been discussed completely, and the existence conditions for sliding segment, sliding mode dynamics and different types of equilibria such as regular equilibrium, pseudo-equilibrium, boundary equilibrium and tangent point have been provided. Further, numerical sliding bifurcation analyses show that the proposed Filippov system has rich sliding bifurcations. Especially, the most interesting results are those for the fixed parameter set as the bifurcation parameter varies, the sliding bifurcations occur sequentially: crossing → buckling → real/virtual equilibrium → buckling → crossing. The key factors which affect the selective treatment measures and the threshold value of infected cases for emerging infectious disease have been discussed in more detail.

Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems

In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a planar Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of global sliding bifurcations as the Grazing-Sliding of periodic orbits and the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction. &nbsp Roughly speaking, we see that locally, and also globally, the regularization of the bifurcations preserve the topological features of the sliding ones.

The equilibrium stability for a smooth and discontinuous oscillator with dry friction

In this paper, we investigate the equilibrium stability of a Filippov-type system having multiple stick regions based upon a smooth and discontinuous (SD) oscillator with dry friction. The sets of equilibrium states of the system are analyzed together with Coulomb friction conditions in both $$(f_{\mathrm{n}}, f_{\mathrm{s}})$$ and $$(x, {\dot{x}})$$ planes. In the stability analysis, Lyapunov functions are constructed to derive the instability for the equilibrium set of the hyperbolic type and LaSalle’s invariance principle is employed to obtain the stability of the non-hyperbolic type. Analysis demonstrates the existence of a thick stable manifold and the interior stability of the hyperbolic equilibrium set due to the attractive sliding mode of the Filippov property, and also shows that the unstable manifolds of the hyperbolic-type are that of the endpoints with their saddle property. Numerical calculations show a good agreement with the theoretical analysis and an excellent efficiency of the approach for equilibrium states in this particular Filippov system. Furthermore, the equilibrium bifurcations are presented to demonstrate the transition between the smooth and the discontinuous regimes.

Rigorous numerics for piecewise-smooth systems: A functional analytic approach based on Chebyshev series

In this paper, a rigorous computational method to compute solutions of piecewise-smooth systems using a functional analytic approach based on Chebyshev series is introduced. A general theory, based on the radii polynomial approach, is proposed to compute crossing periodic orbits for continuous and discontinuous (Filippov) piecewise-smooth systems. Explicit analytic estimates to carry the computer-assisted proofs are presented. The method is applied to prove existence of crossing periodic orbits in a model nonlinear Filippov system and in the Chua’s circuit system. A general formulation to compute rigorously crossing connecting orbits for piecewise-smooth systems is also introduced.

A Dynamical Analysis of a Piecewise Smooth Pest Control SI Model

In this paper, we propose a piecewise smooth SI pest control system to model the process of spraying pesticides and releasing infectious pests. We assume that the pest population consists of susceptible pests and infectious pests, and that the disease spreads horizontally between pests. We take the susceptible pest as the control index on whether to implement chemical control and biological control strategies. Based on the theory of Filippov system, the sliding-mode domain and conditions for the existence of real equilibria, virtual equilibria, pseudo-equilibrium and boundary equilibria are given. Further, we show the global stability of real equilibria (or boundary equilibria) and pseudo-equilibrium. Our results can provide theoretical guidance for the problem of pest control.

Stochastic Perturbations of Periodic Orbits with Sliding

Vector fields that are discontinuous on codimension-one surfaces are known as Filippov systems and can have attracting periodic orbits involving segments that are contained on a discontinuity surface of the vector field. In this paper we consider the addition of small noise to a general Filippov system and study the resulting stochastic dynamics near such a periodic orbit. Since a straight-forward asymptotic expansion in terms of the noise amplitude is not possible due to the presence of discontinuity surfaces, in order to quantitatively determine the basic statistical properties of the dynamics, we treat different parts of the periodic orbit separately. Dynamics distant from discontinuity surfaces is analyzed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analyzed through stochastic averaging methods. The influence of noise on points at which the periodic orbit escapes a discontinuity surface is determined by zooming into the transition point. We combine the results to quantitatively determine the effect of noise on the oscillation time for a three-dimensional canonical model of relay control. For some parameter values of this model, small noise induces a significantly large reduction in the average oscillation time. By interpreting our results geometrically, we are able to identify four features of the relay control system that contribute to this phenomenon.

Piecewise virus-immune dynamic model with HIV-1 RNA-guided therapy

Clinical studies have used CD4 T cell counts to evaluate the safety or risk of plasma HIV-1 RNA-guided structured treatment interruptions (STIs), aimed at maintaining CD4 T cell counts above a safe level and plasma HIV-1 RNA below a certain level. However, quantifying and evaluating the impact of STIs on the control of HIV replication and on activation of the immune response remains challenging. Here we extend the virus-immune dynamic system by including a piecewise smooth function to describe the elimination of HIV viral loads and the activation of effector cells under plasma HIV-1 RNA-guided therapy, in order to quantitatively explore the STI strategies. We theoretically investigate the global dynamics of the proposed Filippov system. Our main results indicate that HIV viral loads could either go to infinity or be maintained below a certain level or stabilize at a previously given level, depending on the threshold level and initial HIV virus loads and effector cell counts. This suggests that proper combinations of threshold and initial HIV virus loads and effector cell counts, based on threshold policy, can successfully preclude exceptionally high growth of HIV virus and, in particular, maximize the controllable region.

A Universal Feedback Controller for Discontinuous Dynamical Systems Using Nonsmooth Control Lyapunov Functions

The consideration of nonsmooth Lyapunov functions for proving stability of feedback discontinuous systems is an important extension to classical stability theory since there exist nonsmooth dynamical systems whose equilibria cannot be proved to be stable using standard continuously differentiable Lyapunov function theory. For dynamical systems with continuously differentiable flows, the concept of smooth control Lyapunov functions was developed by Artstein to show the existence of a feedback stabilizing controller. A constructive feedback control law based on a universal construction of smooth control Lyapunov functions was given by Sontag. Even though a stabilizing continuous feedback controller guarantees the existence of a smooth control Lyapunov function, many systems that possess smooth control Lyapunov functions do not necessarily admit a continuous stabilizing feedback controller. However, the existence of a control Lyapunov function allows for the design of a stabilizing feedback controller that admits Filippov and Krasovskii closed-loop system solutions. In this paper, we develop a constructive feedback control law for discontinuous dynamical systems based on the existence of a nonsmooth control Lyapunov function defined in the sense of generalized Clarke gradients and set-valued Lie derivatives.

A Filippov system describing the effect of prey refuge use on a ratio-dependent predator–prey model

In this paper, a Filippov ratio-dependent predator–prey model is proposed to describe the effect on behavioral refuges caused by prey instinct anti-predator behavior. The proposed model extends the classical ratio-dependent predator–prey model by combining a prey–predator ratio that describes the behavioral refuges make sense once it is less than a certain threshold. One of the prominent mathematical features of our model, distinguishing from the classical one, is that there exist singular points on discontinuous surface whose characteristics determine the main dynamical behaviors of this model. The complete analysis of topological structures of orbits near all the singular points is presented. We show 14 types of system behaviors realized for various parameter values. In particular, globally stable pseudo-equilibrium and globally finite time stable canard cycle are shown to exist in some ranges of parameter values, which cannot be achieved in classical model.

Codimension-1 Sliding Bifurcations of a Filippov Pest Growth Model with Threshold Policy

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.

Existence of multiple sliding segments and bifurcation analysis of Filippov prey–predator model

In a prey–predator model, an effective management strategy called the threshold policy control (TPC) is proposed, resulting in Filippov system. Considering the limitation of environmental resources, Filippov system with the weighted sum as the index is put forward in the prey–predator model with Ivlev’s function response. Firstly, the existences of sliding segments in four cases have been discussed completely. Further, the sliding mode dynamics, the existences of different types of equilibria and tangent point, regular/virtual equilibrium and sliding mode bifurcations have been addressed. Moreover, the results obtained in present work indicate that the local sliding bifurcations such as boundary focus, boundary node and tangent bifurcations occur sequentially with the threshold value varying. Finally, some global sliding bifurcations including touching and buckling bifurcations are investigated by employing numerical techniques.

Numeric-analytic solutions of the smooth and discontinuous oscillator

Abstract Earlier works on the smooth and discontinuous (SD) oscillator concentrated mainly on the time domain analysis using analytical, semi-analytical and numerical integration methods. In this paper, the frequency domain analysis of the SD oscillator subjected to harmonic excitation which is as important and giving further insight into the dynamics is carried out. Multi-Harmonic Balance Method (MHBM) in combination with arc length continuation is used to obtain the periodic solutions and their branches in the frequency domain for different values of the smoothing parameter α and exciting frequency ω. Stability of the periodic motions and bifurcation behavior are analyzed using the Floquet theory. For the discontinuous case, the oscillator is treated as a Filippov system and an event driven numerical integration method is used to obtain the response. For α > 1 , the dynamics of the SD oscillator is similar to that of the hardening Duffing oscillator, for α = 1 , it is like that of the Ueda oscillator and for 0 α 1 it is like that of the Duffing oscillator with double well potential. The SD oscillator exhibits period 1 solutions, higher order periodic solutions, chaotic solutions through symmetry breaking bifurcations, period doubling and boundary crises in different parameter ranges. Chaos is observed over a larger frequency range interspersed by narrow windows of higher order periodic solutions.

Three crossing limit cycles in planar piecewise linear systems with saddle-focus type

This paper presents an analysis on the appearance of limit cycles in planar Filippov system with two linear subsystems separated by a straight line. Under the restriction that the orbits with points in the sliding and escaping regions are not considered, we provide firstly a topologically equivalent canonical form of saddle-focus dynamic with five parameters by using some convenient transformations of variables and parameters. Then, based on a very available fourth-order series expansion of the return map near an invisible parabolic type tangency point, we show that three crossing limit cycles surrounding the sliding set can be bifurcated from generic codimensionthree singularities of planar discontinuous saddle-focus system. Our work improves and extends some existing results of other researchers.

Mathematical analysis of a model for the growth of the bovine corpus luteum

The corpus luteum (CL) is an ovarian tissue that grows in the wound space created by follicular rupture. It produces the progesterone needed in the uterus to maintain pregnancy. Rapid growth of the CL and progesterone transport to the uterus require angiogenesis, the creation of new blood vessels from pre-existing ones, a process which is regulated by proteins that include fibroblast growth factor 2 (FGF2). In this paper we develop a system of time-dependent ordinary differential equations to model CL growth. The dependent variables represent FGF2, endothelial cells (ECs), luteal cells, and stromal cells (like pericytes), by assuming that the CL volume is a continuum of the three cell types. We assume that if the CL volume exceeds that of the ovulated follicle, then growth is inhibited. This threshold volume partitions the system dynamics into two regimes, so that the model may be classified as a Filippov (piecewise smooth) system. We show that normal CL growth requires an appropriate balance between the growth rates of luteal and stromal cells. We investigate how angiogenesis influences CL growth by considering how the system dynamics depend on the dimensionless EC proliferation rate, ρ₅. We find that weak (low ρ₅) or strong (high ρ₅) angiogenesis leads to 'pathological' CL growth, since the loss of CL constituents compromises progesterone production or delivery. However, for intermediate values of ρ₅, normal CL growth is predicted. The implications of these results for cow fertility are also discussed. For example, inadequate angiogenesis has been linked to infertility in dairy cows.

A Filippov system describing media effects on the spread of infectious diseases

A Filippov epidemic model with media coverage is proposed to describe the real characteristics of media/psychological impact in the spread of an infectious disease. We extend the existing models by incorporating a piecewise continuous transmission rate to describe that the media coverage exhibits its effect once the number of infected individuals exceeds a certain critical level. Mathematical and bifurcation analyses with regard to the local, global stability of equilibria and local sliding bifurcations are performed. Our main results show that the system stabilizes at either the equilibrium points of the two subsystems or the new endemic state induced by the on–off media effect, depending on the threshold levels. The finding suggests that a previously chosen level of the desired number of infected individuals can be reached when the threshold policy and other parameters are chosen properly.