Non-standard Bifurcations Research Articles

Additional Food Causes Predators to Explode — Unless the Predators Compete

The literature posits that an introduced predator population is able to drive its target pest population to extinction, if supplemented with high quality additional food of sufficient quantity. We show this approach actually leads to infinite time blow-up of the predator population, so is unpragmatic as a pest management strategy. We propose an alternate model in which the additional food induces predator competition. Analysis of this model indicates that depending on the competition parameter [Formula: see text], one can have global stability of the pest-free state, bistability dynamics, or up to three interior equilibria. As [Formula: see text] and the additional food quantity [Formula: see text] are varied standard codimension one and codimension two bifurcations are observed. We also use structural symmetries to construct several nonstandard bifurcations such as saddle-node-transcritical bifurcation (SNTC) in codimension two and a cusp-transcritical bifurcation (CPTC), also in codimension two. We further use symmetry to construct a novel pitchfork-transcritical bifurcation (PTC) in codimension two, thus explicitly characterizing a new organizing center of the model. Our findings indicate that increasing additional food in predator–pest models can hinder bio-control, contrarily to some of the literature. However, additional food that also induces predator competition, leads to rich dynamics and enhances bio-control.

Dynamical Analysis of Border Collision Bifurcation in One-Cycle Controlled Single-Inductor Dual-Output Boost DC-DC Converters

In this paper, the evolution of three types of border collision bifurcations are investigated thoroughly for a high-order single-inductor double-output DC-DC converter with one-cycle control (OCC). Multi-piecewise inductor current with more than one border and cross-coupling effect between different output ports are prominent features of the system, resulting in unexpected period incrementing cascade and transition from periodicity to chaos as a parameter is varied. A high-order nonlinear iterative map model is proposed, which reveals the underlying mechanisms of those complicated nonstandard and unsmooth bifurcations in the multiple-output system. Additionally, theoretical analysis shows that those nonstandard bifurcations arise from the boundary collisions, and more specifically, a fixed point suddenly appears in the borderline between various regions. Besides eigenvalue loci changes abruptly at the bifurcation point, the characteristic of border collision bifurcation is the topology sequence changes. Furthermore, the parameter boundaries are given to provide useful guidance for design. Finally, PSpice circuit experiments are performed to verify the above theoretical and numerical results.

Positive solutions of a nonlocal singular elliptic equation by means of a non-standard bifurcation theory

We consider a nonlocal elliptic equation arising in a prey–predator model whose nonlocal term is singular. We use the Leray–Schauder degree to prove the existence of an unbounded continuum of positive solutions emanating from the trivial solution. As application, we study nonlocal and singular elliptic equations of the type logistic and Holling–Tanner.

Diversity and transition characteristics of sticking and non-sticking periodic impact motions of periodically forced impact systems with large dissipation

A forced vibration system with dual bodies and an endstop is considered in this paper. A large energy loss is considered as one of the mass blocks of the system hits the endstop, and the impact associated with the large energy loss is first assumed to be completely plastic. The plastic impact may bring about the occurrence of the sticking phase, which is equivalent to a change in the structure of the forced vibration system at a certain stage after the impact. The incidence relation between dynamical characteristics and model parameters is studied through the multi-target and multi-parameter collaborative simulation analysis for determining the reasonable matching range of parameters. Pattern types, occurrence regions, distribution regularities and bifurcation characteristics of periodic and subharmonic impact vibrations are presented on a series of parameter planes. The key features of Poincare mapping, associated with the plastic impacts, are primarily manifested in piecewise continuity caused by sliding bifurcation and grazing discontinuity induced by grazing bifurcation. Integrative effects of these two nonstandard bifurcations can bring about some abnormal transitions to occur. The large dissipation case associated with small collision recovery coefficient is briefly analyzed, and the induced mechanism of chatter-sticking motion and the incidence relation between the sticking characteristics and the restitution coefficient R are discussed. The nonstandard dynamic characteristics associated with the plastic impacts are further demonstrated by dynamic mechanical behaviors of two practical impact machines applied in engineering.

Nonstandard transitions in the Kuramoto model: a role of asymmetry in natural frequency distributions

We study transitions in the Kuramoto model by shedding light on asymmetry in the natural frequency distribution, which has been assumed to be symmetric in many previous studies. The asymmetry brings two nonstandard bifurcation diagrams, with the aid of bimodality. The first diagram consists of stationary states, and has the standard continuous synchronization transition and a subsequent discontinuous transition as the coupling strength increases. Such a bifurcation diagram has been also reported in a variant model, which breaks the odd symmetry of the coupling function by introducing the phase lag. The second diagram includes the oscillatory state emerging from the partially synchronized state and followed by a discontinuous transition. This diagram is firstly revealed in this study. The two bifurcation diagrams are obtained by employing the Ott–Antonsen ansatz, and are verified by direct N-body simulations. We conclude that the asymmetry in distribution, with the bimodality, plays a similar role to the phase lag, and diversifies the transitions.

Hidden and Nonstandard Bifurcation Diagram of an Alternate Quadratic System

Alternate quadratic systems [Formula: see text] where [Formula: see text] and [Formula: see text] are different parameters, seem to be interval maps in a range of the parameter values. However, after a careful graphical analysis of their bifurcation diagrams we conclude that this is true only for system B, but not for system A. In system A we find a hidden and nonstandard bifurcation diagram (“hidden” because it is not visible at normal resolution and “nonstandard” because the bifurcation diagram is empty for some ranges of the parameter values). The different behavior of the underlying critical polynomial in the range of parameter values in both alternate quadratic systems explains why the hidden and nonstandard bifurcation diagram is present in system A and not in system B. The analysis of the Lyapunov exponent also shows both the existence and the different behavior of the hidden bifurcation diagram of system A.

Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.

Dynamics of vibro-impact mechanical systems with large dissipation

An n-degree-of-freedom system having placed single stop and subjected to periodic excitation is considered. Based on the analysis of dynamics of the vibratory system with plastic impacts, we introduce a (2 n−1)-dimensional map with dynamical variables defined at the impact instants. The nonlinear dynamics of the vibro-impact system is analyzed by using the Poincaré map, in which piecewise property and singularity are found to exist. The piecewise property is caused by the transitions of free flight and sticking motions of the impact mass immediately after the impact, and the singularity of the map is generated via the grazing contact of both the impact mass and the rigid stop and corresponding instability of periodic-impact motions. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The single-impact periodic motions and disturbed map, associated with free flight motion of the system, are derived analytically. Stability, sliding and period-doubling bifurcations of the single-impact periodic motions are analyzed by the presentation of results for a three-degree-of-freedom plastic impact oscillator. Finally two actual examples, the impact-forming machine and inertial shaker, are considered to further analyze periodic-impact motions and bifurcations of plastic impact oscillators. The free flight and sticking solutions of two impact machines are analyzed numerically, and regions of existence and stability of different periodic-impact motions are therefore presented. The influence of non-standard bifurcations and system parameters on dynamics of the vibro-impact machines is elucidated accordingly.

Periodic motions and bifurcations of a vibro-impact system

Based on the analysis of a two-degree-of-freedom plastic impact oscillator, we introduce a three-dimensional map with dynamical variables defined at the impact instants. The non-linear dynamics of the vibro-impact system is analyzed by using the Poincaré map, in which piecewise property and singularity are found to exist. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of map is generated via the grazing contact of two masses and corresponding instability of periodic motions. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. Simulations of the free flight and sticking solutions are carried out, and regions of existence and stability of different impact motions are therefore presented in ( δ, ω) plane of dimensionless clearance δ and frequency ω. The influence of non-standard bifurcations on dynamics of the vibro-impact system is elucidated accordingly.

Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics

Analytical and experimental investigations are performed in order to characterize the dynamic behaviour of a non-smooth rotational oscillator, which exhibits multiple discontinuity boundaries in the phase space. The physical system consists of a rotating body subjected to an elastic restoring force and in contact with one or two rough discs rotating with constant driving velocities. The presence of multiple discontinuity boundaries caused by frictional contacts leads to non-standard bifurcations that are studied by means of a simple mechanical model.

PERIODIC MOTIONS AND BIFURCATIONS OF VIBRATORY SYSTEMS WITH PLASTIC IMPACTS REPEATED

Vibratory systems with repeated impacts are considered. Dynamics of such systems, in inelastic impact cases, are studied with special attention to existence of two different types of periodic-impact motions, bifurcations and singularity by applying bifurcation theory of mapping. Regularity and transition of two types of periodic-impact motions are studied by use of a mapping derived from the equations of motion. The mapping of vibratory systems with repeated inelastic impacts is of piecewise property due to synchronous and non-synchronous motions of impact components immediately after the impact, and singularities caused by the grazing contact motions of impact components. The piecewise property and grazing singularity of Poincare mapping of such systems lead to non-standard bifurcations of periodic-impact motions. The influence of the piecewise property and singularities on global bifurcations and transitions to chaos is elucidated. The routes from periodic-impact motions to chaos are analyzed by numerical analyses.

BIFURCATIONS IN HYBRID MECHANICAL SYSTEMS WITH DISCONTINUITY BOUNDARIES

The paper studies the dynamics of a stop-belt friction oscillator (SBO). The system consists of a visco-elastic oscillator which is dragged by a rough support moving with constant driving velocity and can collide with a rigid obstacle. The peculiarity of this system is that, in the phase space, it exhibits two different discontinuity boundaries. Nonsmooth responses are consequently observed with nonstandard attracting properties. The evolution of steady state limit sets as the position of the obstacle is varied is described. The nonsmoothness sets of the system and, in particular, the presence of discontinuity boundaries caused by friction and impacts, lead to nonstandard bifurcations which are studied here by means of analytical and numerical tools.

Nonstandard Bifurcations in Oscillators with Multiple Discontinuity Boundaries

The model of a double-belt friction oscillator is proposed, which exhibitsmultiple discontinuity boundaries in the phase space. The system consists of a simpleoscillator dragged by two different rough supports moving with constant driving velocitiesand subjected to an elastic restoring force and viscous damping. Self-sustained oscillationshave been observed to occur, with nonstandard attracting properties. By consideringthe problem from a nonsmooth dynamical systems perspective, the evolution ofsteady state attractors as the velocities of the belts are varied is described. The nonsmoothnesssets of the system at hand and, in particular, the presence of multiple discontinuityboundaries, lead to nonstandard bifurcations which are studied here by meansof analytical and numerical tools.

Bifurcations in piecewise-smooth feedback systems

This paper is concerned with the analysis and classification of non-standard bifurcations in piecewise smooth feedback systems. Bifurcations involving fixed points in maps, equilibria and limit cycles in flows are considered with particular attention to: border collisions of fixed points in maps; non-smooth bifurcations of equilibria, grazing bifurcations and sliding bifurcations of limit cycles in flows. The aim is to describe existing and novel results to form the basis of a consistent theory of bifurcations in such systems. In so doing, a novel approach to classify non-smooth bifurcations of equilibria in flows is presented. Experimental results on pulse-width modulated voltage-mode controlled DC/DC power converters motivates the relevance to applications of the analysis proposed.

Dynamics of a Simple Damped Oscillator Undergoing Stick-Slip Vibrations

The dynamics of a simple dynamical system subjected to an elastic restoring force, viscous damping and dry friction forces is investigated. Self-sustained oscillations occur with non-standard attracting properties. Discontinuity of the governing equations leads to non-standard bifurcations, which are studied here, with analytical and numerical tools.