Narrow Results

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.

In this paper, a freight train suspension system is presented for all possible types of motion. The suspension system experiences impacts and friction between wedges and bolster. The impacts cause the chatter motions between wedges and bolster, and the friction will cause the stick and nonstick motions between wedges and bolster. Due to the wedge effect, the suspension system may become stuck and not move, which can cause the suspension to lose functions. To discuss such phenomena in the freight train suspension systems, the theory of discontinuous dynamical systems is used, and the motion mechanism of impacting chatter with stick and stuck is discussed. The analytical conditions for the onset and vanishing of stick motions between the wedges and bolster are presented, and the condition for maintaining stick motion was achieved as well. The analytical conditions are developed for the onset and vanishing conditions for stuck motion. An analytical prediction of periodic motions relative to impacting chatter with stick and stuck motions in the train suspension is performed through the mapping dynamics. The corresponding analyses of local stability and bifurcation are carried out, and the grazing and stick conditions are used to determine periodic motions. Numerical simulations illustrate periodic motions of stick and stuck motions. Finally, from field testing data, the effects of wedge angle on the motions of the suspension are presented to find a more desirable suspension response for design.

In this paper, we study a four-parameters piecewise-smooth dry friction oscillator from Control theory. Using Filippov's convention, we prove the existence of a codimension-1 bifurcation which gives rise to a normally hyperbolic set composed by a family of attracting cylinders. This bifurcation exhibits interesting discontinuous oscillation phenomena. We also present consistent numerical simulations.

In this paper, we consider bifurcations of a class of planar piecewise smooth differential systems constituted by a general linear system and a quadratic Hamiltonian system. The linear system has four parameters. When the parameters vary in different regions, the left linear system can have a saddle, a node or a focus. For each case, we provide a completely qualitative analysis of the dynamical behavior for this piecewise smooth system. Our results generalize and improve the results in this direction.

This paper proposes a novel chaotic jerk system, which is defined on four domains, separated by codimension-2 discontinuity surfaces. The dynamics of the proposed system are conveniently described and analyzed through a generalization of the Poincaré map which is constructed via an explicit solution of each subsystem. This provides an approach to formulate a robust bifurcation problem as a nonlinear inhomogeneous eigenvalue problem. Also, we establish some criteria for the existence of a period-doubling bifurcation and discuss some of the interesting categories of complex behavior such as multiple period-doubling bifurcations and chaotic behavior when the trajectory undergoes a segment of sliding motion. Our results emphasize that the sharp switches in the behavior are mainly responsible for generating new and unique qualitative behavior of a simple linear system as compared to the nonlinear continuous system.

Sliding bifurcations of different types are defined to characterize the topological features of trajectories around the switching manifolds of nonsmooth dynamical systems. In this paper, the stick-slip transitions, which are related to the dynamical scenarios of sliding bifurcations, of the self-excited dry friction backward whirls of a general rotor/stator rubbing system, are investigated. The four-degree-of-freedom piecewise smooth rotor/stator rubbing model is said to be general because it takes into account main factors in the rotor/stator rubbing systems, including both the dynamics of the rotor and the stator as well as the dry friction and the flexibility on the contact surfaces. The switching manifold that separates two discontinuous vector fields is defined as the curved hypersurface in a nine-dimensional extended state space where the relative velocity at the contact points equals zero. After deriving the formulae defining the sliding regions and their boundaries for the piecewise smooth system, two extreme cases with rigid and soft contact surfaces are theoretically analyzed and confirmed to correspond respectively to a continuous pure rolling with full sliding region and a continuous crossing without sliding region on the switching manifold. Furthermore, three types of sliding bifurcations, namely, crossing-sliding, grazing-sliding and switching-sliding, are observed in the dry friction backward whirls of the present model in a semi-analytical way. Moreover, hybrids of the three kinds of the sliding solutions in one period of oscillation are also identified with the variation of system parameters and initial conditions. The main scenarios of the switching transition of sliding bifurcations in the self-excited dry friction backward whirl of the general rotor/stator rubbing system with the variation of system parameters are also summarized.

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.

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.

Chaos in a DC–DC boost converter operating in discontinuous conduction mode when controlled by sliding mode control is detected. A route to chaos through sliding bifurcations is shown along with the coexistence of the chaotic regime with a stable operating point. A bifurcation analysis in the sliding vector field is carried out with the objective of defining a range of values for which control parameters guarantee the robustness of the system, at least locally. The analysis is based on the study of possible sliding bifurcations, that is, bifurcations inherent to the sliding dynamics. It is rigorously shown that there is an unstable limit cycle in the sliding dynamics due to a subcritical Hopf bifurcation. The existence of this limit cycle and its evolution with parameters turn out to be of great importance for the stability and robust control of this type of power converter.

A Filippov–Gause predator–prey model is proposed, which suggests that cooperation and competition among predators depend on prey abundance. In this case, when the prey abundance is high, predators cooperate in hunting, but when the prey abundance decreases and falls below a critical level, predators compete for scarce food. The proposed model is analyzed in terms of the existence and stability of its equilibria and its pseudo-equilibrium, which is located on the differentiable curve separating the two vector fields. Given the conditions found which determine the existence of cycles and limit cycles, contrasted with a bifurcation diagram, we find that the proposed model can have an asymptotically stable limit cycle around the pseudo-equilibrium and two positive inner equilibria, together with an unstable limit cycle around the pseudo-equilibrium. Similarly, if the model has a pseudo-equilibrium and no positive inner equilibria, then we could have at least two limit cycles, one stable and one unstable, surrounding the pseudo-equilibrium.

Teixeira singularity (TS) is the transverse intersection point of invisible quadratic tangency lines in 3D piecewise-smooth systems, which significantly impacts both local and global dynamics. This paper considers the 3D nonlinear Filippov system composed of Yang system and Sprott-C system with a symmetric pair of TSs. In parameters space, this Filippov system exhibits rich bifurcation phenomena, such as the boundary equilibrium bifurcations and a symmetric pair of sliding transcritical bifurcations. By constructing the Poincaré map and analyzing the number of fixed points, we present the conditions for the existence of crossing limit cycles. Furthermore, we observe the phenomenon of compound bifurcation occurring at TS, namely the TS bifurcation. This bifurcation results from the combination of transcritical bifurcation of sliding dynamics and Bogdanov–Takens bifurcation of crossing dynamics. Finally, a new chaotic attractor, which consistently surrounds two TSs, is found through numerical analysis.