Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, there exist a large number of switching dynamical systems that have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. In this paper we present a systematic approach to the problem of analyzing the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We first motivate the analysis by considering the bifurcations occurring in a familiar physical system-the static VAR compensator used in electrical power systems-and then proceed to formulate the theory needed to explain the bifurcation behavior of such systems. We then integrate the observed bifurcation phenomenology of the compensator with the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps.     

Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems

In this paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems) which are everywhere continuous. We consider phase space topology of systems with one-dimensional slow dynamics and one-dimensional fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and the differentiability is lost across the switching surface. In the full system the slow manifold is no longer continuous, and there is an O(?) discontinuity across the switching manifold, but the discontinuity cannot qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncovered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and the presence of a boundary-equilibrium bifurcation of a fold type is shown.     

Degenerate discontinuity-induced bifurcations in tapping-mode atomic-force microscopy

This paper documents the existence of degenerate bifurcation scenarios in the low-contact-velocity dynamics during tapping-mode atomic-force microscopy. Specifically, numerical analysis of a model of the microscope dynamics shows branch point and isola bifurcations associated with the emergence of two families of saddle–node bifurcation points along a branch of low-amplitude oscillations. The paper argues for the origin of the degenerate bifurcations in the existence of a periodic steady-state trajectory that (i) achieves tangential contact with a discontinuity surface in a piecewise smooth model of the cantilever response and (ii) retracts from the surface under variations in either direction along a line segment in parameter space. Specifically, the discontinuity-mapping technique is here rigorously applied to a general situation of such degenerate contact showing the codimension-two nature of these bifurcations for appropriately chosen parameter values. The discontinuity-mapping-based normal form derived here is a novel extension of that derived in Dankowicz and Nordmark (2000) [28] in the case that (ii) does not hold. In addition, the paper includes a quantitative reflection on the relative importance of discontinuities in the attractive and repulsive force components in producing the predicted bifurcations.     

Remerging Feigenbaum trees in dynamical systems

Finite sequences of remerging period-doubling bifurcations have been recently observed in a variety of physically interesting dynamical systems. We show here that such remerging Feigenbaum trees are quite common in models with more than one parameter and discuss a number of criteria under which they are generally observed. These criteria are applied to simple mappings as well as the conservative Duffing's equation where the formation of a primary “bubble” is seen to lead to higher-order bubbles and hence to remerging Feigenbaum sequences. In the case of Duffing's equation, we follow the development of one such sequence, with the aid of the variation of the winding number along a symmetry axis of the problem.     

Bifurcations and selection of equilibria in a simple cosymmetric model of filtrational convection

A three-dimensional set of ordinary differential equations that constitutes a simple abstract model of Darcy convection is investigated. The model reproduces a number of effects that are typical for dynamic systems with nontrivial cosymmetry. Nontrivial cosymmetry can give rise to a continuous family of equilibria where, in this case, the equilibrium stability spectrum varies along the family. The family of equilibria and its stability are examined analytically, and special bifurcations that occur in the system are investigated. It is shown that discrete and continual symmetries, called "flash symmetries," can be present in the system for certain parameter values. Computer experiments on the selection of equilibria in the symmetric and cosymmetric cases have been carried out. They showed that, for initial points that are far enough from a cycle of equilibria, the neighborhood of a single equilibrium is established in the case of cosymmetry, but all the equilibria are equivalent in the case of symmetry. The authors hope that these results, as well as the formulation of the problems and the approach to their solution, will serve as a sample in the investigation of more complex systems in mathematical physics. (c) 1999 American Institute of Physics.     

The phase-modulated logistic map

We study the logistic mapping with the nonlinearity parameter varied through a delayed feedback mechanism. This history dependent modulation through a phaselike variable offers an enhanced possibility for stabilization of periodic dynamics. Study of the system as a function of nonlinearity and modulation parameters reveals new phenomena: In addition to period-doubling and tangent bifurcations, there can be bifurcations where the period increases by unity. These are extensions of crises that arise in nonlinear dynamical systems. Periodic orbits in this system can be systematized via the kneading theory, which in the present case extends the analysis of Metropolis, Stein, and Stein for unimodal maps.     

Interactions between global and grazing bifurcations in an impacting system

It is well known that the locus of boundary crises in smooth systems contains gaps that give rise to periodic windows. We show that this phenomenon can also be observed in an impacting system, and that the mechanism by which these gaps are created is different. Namely, here gaps are created and disappear at points along the branches of boundary crises where they are intersected by branches of grazing bifurcations. We locate a novel type of double-crisis vertex which we call a grazing-crisis vertex. Additionally, we illustrate several types of basin-boundary metamorphosis that are intricately related with grazing bifurcations.     

Grazing and border-collision in piecewise-smooth systems: a unified analytical framework

A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes. This links grazings with border-collisions in nonsmooth maps. Contrary to previous literature, piecewise linear maps correspond only to nonsmooth discontinuity boundaries. All other maps have either square-root or (3/2)-type singularities.     

On creation of Hopf bifurcations in discrete-time nonlinear systems

Bifurcation characteristics of a nonlinear system can be manipulated by small controls. In this paper, we present a control method to create Hopf bifurcations in discrete-time nonlinear systems. The critical conditions for the Hopf bifurcations are discussed. The center manifold method, normal form technique and the Iooss's Hopf bifurcation theory are employed in the derivation of the control gain. Numerical demonstration is provided. (c) 2002 American Institute of Physics.     

Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems

A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional piecewise-smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for piecewise-smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.     

Multi-scale energy exchanges between a nonlinear oscillator of Bouc-Wen type and another coupled nonlinear system

The concept of energy exchange between coupled oscillators can be endowed for wide variety of applications such as control and energy harvesting. It has been proved that by coupling an essential nonlinear oscillator (cubic nonlinearity) to a main system (mostly linear), the latter system can be controlled in a one way and almost irreversible manner. The phenomenon is called energy pumping and the coupled nonlinear system is named as nonlinear energy sink (NES). The process of energy transfer from the main system to the nonlinear smooth or non-smooth attachment at different scales of time can present several scenarios: It can be attracted to periodic behaviors which present low or high energy levels for the main system and/or to quasi-periodic responses of two oscillators by persistent bifurcations between their stable zones. In this paper we analyze multi-scale dynamics of two attached oscillators: a Bouc-Wen type in general (in particular: a Dahl type and a modified hysteresis system) and a NES (nonsmooth and cubic). The system behavior at fast and first slow times scales by detecting its invariant manifold, its fixed points and singularities will be analyzed. Analytical developments will be accompanied by some numerical examples for systems that present quasi-periodic responses. The endowed Bouc-Wen models correspond to the hysteretic behavior of materials or structures. This paper is clearly connected with the dynamics of systems with hysteresis and nonlinear dynamics based energy harvesting.     

A mathematical framework for critical transitions: Bifurcations, fast-slow systems and stochastic dynamics

Bifurcations can cause dynamical systems with slowly varying parameters to transition to far-away attractors. The terms “critical transition” or “tipping point” have been used to describe this situation. Critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance. The main goal of this paper is to give an overview which standard mathematical theories can be applied to critical transitions. We shall focus on early-warning signs that have been suggested to predict critical transitions and point out what mathematical theory can provide in this context. Starting from classical bifurcation theory and incorporating multiple time scale dynamics one can give a detailed analysis of local bifurcations that induce critical transitions. We suggest that the mathematical theory of fast-slow systems provides a natural definition of critical transitions. Since noise often plays a crucial role near critical transitions the next step is to consider stochastic fast-slow systems. The interplay between sample path techniques, partial differential equations and random dynamical systems is highlighted. Each viewpoint provides potential early-warning signs for critical transitions. Since increasing variance has been suggested as an early-warning sign we examine it in the context of normal forms analytically, numerically and geometrically; we also consider autocorrelation numerically. Hence we demonstrate the applicability of early-warning signs for generic models. We end with suggestions for future directions of the theory.     

Fermi acceleration revisited

Mappings that have been used to describe the Fermi acceleration mechanism are examined. It is shown that results which appear to be contradictory are due to differences in the mapping equations. For those mappings that can be locally approximated by the standard mapping, the value of the nonlinear parameter of the standard mapping, for which the last isolating KAM surface exists, can be used to predict the loss of KAM stability with action for the more general mappings. Previous results of the variation in the density distribution in the stochastic region of the phase space, averaged over phases, is shown to be consistent with the ergodic hypothesis. Fine scale structure of the mappings is found to be model dependent. The standard mapping is a member of a class of mappings which retains some KAM trajectories at arbitrarily large nonlinearity. This feature is not generic to a wider class of mappings discussed in this paper. The stability of two-iteration fixed points are discussed in detail, including the bifurcation sequence for one type of mapping.     

Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam

The aim of this paper is to investigate the multi-pulse global bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case. First, the extended Melnikov method for studying the Shilnikov-type multi-pulse homoclinic orbits and chaos in high-dimensional nonlinear systems is briefly introduced in the theoretical frame. Then, this method is utilized to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam. How to employ this method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example. Finally, the results of numerical simulation are given and also show that the Shilnikov-type multi-pulse chaotic motions can occur for the nonlinear non-planar oscillations of the cantilever beam, which verifies the analytical prediction.     

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectrum is considered both numerically and analytically using previous work of Edelman et al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion does not hold, e.g., real random matrices with Gaussian elements with a large positive mean and finite variance. PACS numbers: 05.45.−a, 05.45.Tp, 89.75.−k, 89.75.Fb     

Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator

This paper is concerned with numerical continuation and analytical investigations of sliding bifurcations in Filippov systems. In particular, a methodology developed for the continuation of grazing bifurcations in impacting systems is used to continue sliding bifurcations in Filippov systems. A dry-friction oscillator is investigated from a sliding bifurcations point of view and a complex two-parameter bifurcation diagram of sliding bifurcations is presented. A number of codimension-two sliding bifurcation points that act as organising centres for codimension-one sliding bifurcations are revealed. Two representative codimension-two points are analysed and unfolded, and the analysis is used to explain the dynamics of the dry-friction oscillator in the neighbourhood of these points.     

Voltage interval mappings for activity transitions in neuron models for elliptic bursters

We performed a thorough bifurcation analysis of a mathematical elliptic bursting model, using a computer-assisted reduction to equationless, one-dimensional Poincaré mappings for a voltage interval. Using the interval mappings, we were able to examine in detail the bifurcations that underlie the complex activity transitions between: tonic spiking and bursting, bursting and mixed-mode oscillations, and finally mixed-mode oscillations and quiescence in the FitzHugh–Nagumo–Rinzel model. We illustrate the wealth of information, qualitative and quantitative, that was derived from the Poincaré mappings, for the neuronal models and for similar (electro)chemical systems.     

Robust dangerous border-collision bifurcations in piecewise smooth systems

Physical and computer experiments involving systems describable by piecewise smooth continuous maps that are nondifferentiable on some surface in phase space exhibit novel types of bifurcations in which an attracting fixed point exists before and after the bifurcation. The striking feature of these bifurcations is that they typically lead to "unbounded behavior" of orbits as a system parameter is slowly varied through its bifurcation value. This new type of border-collision bifurcation is fundamental and robust. A method that prevents such "dangerous border-collision bifurcations" is given. These bifurcations may be found in a variety of experiments including circuits.     

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.