The system in which a small rigid ball is bouncing repeatedly on a massive at table vibrating vertically, so-called the bouncing ball system, has been widely studied. Under the assumption that the table is vibrating with a piecewise polynomial function of time, the bifurcation diagram changes qualitatively depending on the order of the polynomial function. We elucidate the mechanism of the difference in the bifurcation diagrams by focusing on the two-period solution. In addition, we derive the approximate curve of the branch close to the period-doubling bifurcation point in the case of the piecewise cubic function of time for the table vibration. We also performed numerical calculation, and we demonstrate that the approximations well reproduce the numerical results.
Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth. Much of our understanding of these cases relies on a reduction to piecewise linearity near the border-collision. We also review a number of codimension-two bifurcations in which nonlinearity is important.
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, many switching dynamical systems have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. The theory for understanding the bifurcation phenomena in such systems is not available yet. In this paper we present the first approach to the problem of analysing and classifying the bifurcation phenomena in t…
In this paper we report some important results that help in analizing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. Proceeding along similar lines, we obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific e…
Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2 - cycles in a corner-type bifurcation.
An equilibrium of a planar, piecewise-$C^1$, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to $λ_L \pm {\rm i} ω_L$ on one side of the discontinuity and $-λ_R \pm {\rm i} ω_R$ on the other, with $λ_L, λ_R >0$, and the quantity $ Λ= λ_L / ω_L -λ_R / ω_R $ is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the c…
We study dynamics of a ball moving in gravitational field and colliding with a moving table. The motion of the limiter is assumed as periodic with piecewise constant velocity - it is assumed that the table moves up with a constant velocity and then moves down with another constant velocity. The Poincare map, describing evolution from an impact to the next impact, is derived and scenarios of transition to chaotic dynamics are investigated analytically.
We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of bifurcations and chaos associated with it as a function of the delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. Significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for…
We have studied a dissipative version of a one-dimensional Fermi accelerator model. The dynamics of the model is described in terms of a two-dimensional, nonlinear area-contracting map. The dissipation is introduced via innelastic collisions of the particle with the walls and we consider the dynamics in the regime of high dissipation. For such a regime, the model exhibits a route to chaos known as period doubling and we obtain a constant along the bifurcations so called the Feigenbaum's number $δ$.
Recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps which can not be classified among the generic cases like saddle-node, pitchfork or Hopf bifurcations occurring in smooth maps. In this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision. We then present a systematic analysis of such bifurcations by deriving a normal form --- the piecewise linear approximation in the neighbo…
There exists a variety of physically interesting situations described by continuous maps that are nondifferentiable on some surface in phase space. Such systems exhibit novel types of bifurcations in which multiple coexisting attractors can be created simultaneously. The striking feature of these bifurcations is that they lead to fundamentally unpredictable behavior of orbits as a system parameter is varied slowly through its bifurcation value. This unpredictability gradually disappears as the speed of variation of the system parameter through the bifurcation is reduced to zero.
Motivated by bouncing motion of an inelastic particle on a vibrating board, a simple two-dimensional map is constructed and its behavior is studied numerically. In addition to the typical route to chaos through a periodic doubling bifurcation, we found peculiar behavior in the parameter region where two stable periodic attractors coexist. A typical orbit in the region goes through chaotic motion for an extended transient period before it converges into one of the two periodic attractors. The basin structure in this parameter region is almost riddling and the fractal dimension of the basin boun…
Experiments observing the liquid surface in a vertically oscillating container have indicated that modeling the dynamics of such systems require maps that admit states at infinity. In this paper we investigate the bifurcations in such a map. We show that though such maps in general fall in the category of piecewise smooth maps, the mechanisms of bifurcations are quite different from those in other piecewise smooth maps. We obtain the conditions of occurrence of infinite states, and show that periodic orbits containing such states are superstable. We observe period-adding cascade in this system…
Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifu…
In this paper, a memristive Murali-Lakshmanan-Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three segment piecewise linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuty boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity induced non-smooth bifurcations, occur at these boundaries and govern the dynamics of the circu…
From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps
A multistable system generated by a Piecewise Linear (PWL) system based on the jerky equation is presented. The systems behaviour is characterised by means of the Nearest Integer or round(x) function to control the switching events and to locate the corresponding equilibria among each of the commutation surfaces. These surfaces are generated by means of the switching function dividing the space in regions equally distributed along one axis. The trajectory of this type of system is governed by the eigenspectra of the coefficient matrix which can be adjusted by means of a bifurcation parameter. …
In a set of experiments, Couder et. al. demonstrate that an oscillating fluid bed may propagate a bouncing droplet through the guidance of the surface waves. We present a dynamical systems model, in the form of an iterative map, for a droplet on an oscillating bath. We examine the droplet bifurcation from bouncing to walking, and prescribe general requirements for the surface wave to support stable walking states. We show that in addition to walking, there is a region of large forcing that may support the chaotic bouncing of the droplet. Using the map, we then investigate the droplet trajector…
We discuss the bifurcation structure of homoclinic orbits in bimodal one dimensional maps. The universal structure of these bifurcations with singular bifurcation points and the web of bifurcation lines through the parameter space are described. The bifurcations depend on two parameters (codimension 2 bifurcations). We find the bifurcation lines exactly in a symbolic dynamics parameter plane and numerically in the parameter planes of a polynomial map and a piecewise linear map.
This paper provides a first example of constructing Lyapunov functions in a class of piecewise linear systems with limit cycles. The method of construction helps analyze and control complex oscillating systems through novel geometric means. Special attention is stressed upon a problem not formerly solved: to impose consistent boundary conditions on the Lyapunov function in each linear region. By successfully solving the problem, the authors construct continuous Lyapunov functions in the whole state space. It is further demonstrated that the Lyapunov functions constructed explain for the differ…
We show that a simple piecewise-linear system with time delay and periodic forcing gives rise to a rich bifurcation structure of torus bifurcations and Arnold tongues, as well as multistability across a significant portion of the parameter space. The simplicity of our model enables us to study the dynamical features analytically. Specifically, these features are explained in terms of border-collision bifurcations of an associated Poincaré map. Given that time delay and periodic forcing are common ingredients in mathematical models, this analysis provides widely applicable insight.
We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the form…
The paper examines the discrete-time dynamics of neuron models (of excitatory and inhibitory types) with piecewise linear activation functions, which are connected in a network. The properties of a pair of neurons (one excitatory and the other inhibitory) connected with each other, is studied in detail. Even such a simple system shows a rich variety of behavior, including high-period oscillations and chaos. Border-collision bifurcations and multifractal fragmentation of the phase space is also observed for a range of parameter values. Extension of the model to a larger number of neurons is sug…
