Filippov Type Research Articles

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.

Master stability function for piecewise smooth Filippov networks

We consider a network of identical piecewise smooth bimodal systems, also known as systems of Filippov type, that synchronizes along the asymptotically stable periodic orbit of a single agent. We explicitly characterize the fundamental matrix solution of the network along the synchronous solution and extend the Master Stability Function tool to the present case of non-smooth dynamics of Filippov type.

A Filippov type predator prey system with selective harvesting of predator

In this paper, piece-wise smooth predator prey Filippov system to describe the role of refuge to prey and additional food to predators is explored. Logistic growth of prey and Holling type II predation is considered. Selective proportional harvesting of predators is included. Refuge is provided to a fraction of preys when their density is below a threshold value. This reduces the food availability for predators. To compensate this, additional food is provided to predators. The dynamics on the boundary is computed using the Filippov regularization method. The stability and existence of different equilibrium states of the independent smooth subsystems are analyzed. The role of quality of additional food with respect to prey in the characterization of the equilibrium points as regular or virtual equilibrium points for the Filippov system is investigated. It is verified that if only refuge is provided to prey with no additional food to predators then both the interior equilibrium state will never be regular simultaneously for the Filippov system. It is observed from the sliding mode dynamics that there is no escaping region. It is proved that both refuge to prey and additional food to predators enhances the sliding region on the boundary. The analysis of boundary equilibrium points, tangent points and pseudoequilibrium points on the boundary is carried out. The role of effort on harvesting on the expansion or shrinking of the limit cycle is elaborated numerically. Boundary bifurcation for the interior equilibrium state of the subsystems is demonstrated. Numerical simulation is used to elaborate the dynamics of Filippov system.

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.

On some evolution inclusions in non separable Banach spaces

We study a Cauchy problem of a class of nonconvex second-order integro-differential inclusions and a boundary value problem associated to a semilinear evolution inclusion defined by nonlocal conditions in non-separable Banach spaces. The existence of mild solutions is established under Filippov type assumptions.

Bursting Oscillations and Sliding Motion in Permanent Magnet Synchronous Motor Systems with Friction Factor

The main purpose of this paper is to investigate the qualitative effects of external excitation and friction factor on the response of permanent magnet synchronous motor (PMSM) system. Three different modes of bursting oscillations are found. In particular, the introduction of friction function changes the governing equations from a smooth type to a nonsmooth (Filippov) type in which the special sliding motion is observed. The mechanism, attractor structure, vector field structure, and analytic bifurcation conditions of bursting oscillation and sliding motion are discussed in detail. The validity of theoretical results obtained is verified by numerical simulations and analysis.

Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications

The Hβ-Hausdorff–Pompeiu b-metric for β∈[0,1] is introduced as a new variant of the Hausdorff–Pompeiu b-metric H. Various types of multi-valued Hβ-contractions are introduced and fixed point theorems are proved for such contractions in a b-metric space. The multi-valued Nadler contraction, Czervik contraction, q-quasi contraction, Hardy Rogers contraction, weak quasi contraction and Ciric contraction existing in literature are all one or the other type of multi-valued Hβ-contraction but the converse is not necessarily true. Proper examples are given in support of our claim. As applications of our results, we have proved the existence of a unique multi-valued fractal of an iterated multifunction system defined on a b-metric space and an existence theorem of Filippov type for an integral inclusion problem by introducing a generalized norm on the space of selections of the multifunction.

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.

Lyapunov exponents for discontinuous dynamical systems of Filippov type

‎The area of discontinuous dynamical systems is almost a young research area, and the enthusiasm and necessity for analysing these systems have been growing‎. ‎On the other hand‎, ‎chaos appears in a rather wide class of discontinuous systems‎. ‎One of the most important properties of chaos is sensitive dependence on initial conditions‎. ‎Also,‎ the most effective way to diagnosis chaotic systems is defining Lyapunov exponents of these systems‎. ‎In addition‎, ‎defining and calculating Lyapunov exponents for all discontinuous systems are real challenges‎. ‎This paper is devoted to define Lyapunov exponents for discontinuous dynamical systems of Filippov type in order to investigate chaos for these systems‎.

Study of Soft X-ray Emission from Sahand as Filippov Type Plasma Focus Using Different Cathode Type

Study of Soft X-ray Emission from Sahand as Filippov Type Plasma Focus Using Different Cathode Type

Some qualitative properties of mild solutions of a second-order integro-differential inclusion

We prove the Lipschitz dependence on the initial data of the solution set of a Cauchy problem associated to a second-order integro-differential inclusion by using the contraction principle in the space of selections of the multifunction instead of the space of solutions. A Filippov type existence theorem for this problem is also provided.

Two-Parameter Boundary Equilibrium Bifurcations in 3D-Filippov Systems

This work concerns the analysis of boundary equilibrium bifurcations (BEBs) in 3D piecewise-smooth systems of Filippov type. In particular, we consider a family whose vector fields are linear on both sides of the switching boundary and which exhibit two parallel tangency lines each containing a cusp point. This configuration is observed in piecewise-linear control systems in which the control action is discontinuous such as the sliding mode control. We consider a general system of this class and then derive a canonical form to reduce the number of system parameters. The general objective in this work is, from the canonical form, to perform an analysis of the equilibria, stability, sliding dynamics and BEBs. The main result is the classification of the BEBs and its unfoldings in the sliding vector field. This and others results obtained on the existence and stability of equilibria are applied in a practical example involving the control of DC–DC buck power converter.

Nonlinear Dynamic Modeling and Analysis of Self-Oscillating H-Bridge Parallel Resonant Converter Under Zero Current Switching Control: Unveiling Coexistence of Attractors

This paper deals with the global dynamical analysis of an H-bridge parallel resonant converter under a zero current switching control. Due to the discontinuity of the vector field in this system, sliding dynamics may take place. Here, the sliding set is found to be an escaping region. Different tools are combined for studying the stability of oscillations of the system. The desired crossing limit cycles are computed by solving their initial value problem and their stability analysis is performed using Floquet theory. The resulting monodromy matrix reveals that these cycles are created according to a smooth cyclic-fold bifurcation. Under parameter variation, an unstable symmetric crossing limit cycle undergoes a crossing-sliding bifurcation leading to the creation of a symmetric unstable sliding limit cycle. Finally, this limit cycle undergoes a double homoclinic connection giving rise to two different unstable asymmetric sliding limit cycles. The analysis is performed using a piecewise-smooth dynamical model of a Filippov type. Sliding limit cycles divide the state plane in three basins of attraction, and hence, different steady-state solutions may coexist which may lead the system to start-up problems. Numerical simulations corroborate the theoretical predictions, which have been experimentally validated.

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

Several models in the applied sciences are characterized by instantaneous changes in the solutions or discontinuities in the vector field. Knowledge of the geometry of interaction of the flow with the discontinuities can give new insights on the behaviour of such systems. Here, we focus on the class of the piecewise smooth systems of Filippov type. We describe some numerical techniques to locate crossing and sliding regions on the discontinuity surface, to compute the sets of attraction of these regions together with the mathematical form of the separatrices of such sets. Some numerical tests will illustrate our approach.

A note on the solutions of a second-order evolution inclusion in non separable Banach spaces

We consider a Cauchy problem associated to a second-order evolution inclusion in non separable Banach spaces under Filippov type assumptions and we prove the existence of mild solutions.

Sahand Plasma Focus Emitted More Than 35 J in Yield Neon Soft X-ray

X-ray emission from neon plasma produced in Sahand Filippov type plasma focus device is investigated. Detecting system used in our experiments is a five channel pin diode detector. Soft X-ray emissions from neon gas at different charging voltages and working gas pressures are studied and optimum condition for production of soft X-ray for Sahand plasma focus facility is obtained. Results show that for every working pressure there is a charging voltage at which the average soft X-ray yield is maximum and this optimum charging voltage increases with increasing gas pressure. For 0.25, 0.50 and 0.75 Torr neon the optimum voltages are 12, 13 and 16 kV respectively. The highest average soft X-ray yields produced at 0.50 Torr which is about 35.87 ± 1.18 J. Also soft X-ray production decreased after a certain pressure which for our experiments is 0.50 Torr. The variation of relative hard X-ray yield with charging voltage and working gas pressure is also investigated. The results show that 0.50 Torr is also the best operating gas pressure for optimum hard X-ray production in Sahand Filippov type plasma focus device.

On the Solutions of Some Boundary Value Problems for Integro-differential Inclusions of Fractional Order

We study the existence of solutions for fractional integro-differential inclusions with nonlocal boundary conditions and with multi-order fractional integral conditions. We establish Filippov type existence results in the case of nonconvex set-valued maps.

A relaxation theorem for a differential inclusion with “maxima”

Abstract We consider a Cauchy problem associated to a nonconvex differential inclusion with “maxima” and we prove a Filippov type existence result. This result allows to obtain a relaxation theorem for the problem considered.

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.

Bifurcation of periodic orbits in discontinuous systems

This paper treads discontinuous bifurcation in piecewise smooth systems of Filippov type. These bifurcations occur when a fixed point or a periodic orbit crosses with the border between two regions of smooth behavior. A detailed analysis of generalization Poincare map and monodromy matrix which are related shows that subfamily of system with invariant cone-like objects is foliated by periodic orbits and determines its stability. In addition, we introduce a theoretical framework for analyzing 3D perturbed nonlinear piecewise smooth systems and give necessary conditions so that different types of bifurcations occur. The analysis identifies criteria for the existence of a novel bifurcation based on sensitively the location of the return map. Moreover, the piecewise smooth Melnikov function and sufficient conditions of the existence of the periodic orbits for nonlinear perturbed system are explicitly obtained.