Basin Boundary Metamorphoses Research Articles

Using the basin entropy to explore bifurcations

Bifurcation theory is the usual analytic approach to study the parameter space of a dynamical system. Despite the great power of prediction of these techniques, fundamental limitations appear during the study of a given problem. Nonlinear dynamical systems often hide their secrets and the ultimate resource is the numerical simulation of the equations. This paper presents a method to explore bifurcations by using the basin entropy. This measure of the unpredictability can detect transformations of phase space structures as a parameter evolves. We present several examples where the bifurcations in the parameter space have a quantitative effect on the basin entropy. Moreover, some transformations, such as the basin boundary metamorphoses, can be identified with the basin entropy but are not reflected in the bifurcation diagram. The correct interpretation of the basin entropy plotted as a parameter extends the numerical exploration of dynamical systems.

Hidden attractors and metamorphoses of basin boundaries in optomechanics

Hidden attractors and metamorphoses of basin boundaries in optomechanics

The basin boundary of the breakup channel in chaotic rearrangement scattering

We present an example of demonstration for the basin boundaries in classical rearrangement scattering with particular emphasis on the breakup channel. Whereas the basin boundaries of the other arrangement channels are given by stable manifolds of periodic orbits in the interaction region, the basin boundary of the breakup channel is given by the stable manifold of a particular subset in the set of final asymptotes. The geometry of this boundary surface is presented in detail. Further, we discuss the transition to chaos at the energetic threshold of the breakup channel and the related basin boundary metamorphosis.

Global dynamics of the dry friction oscillator with shape memory alloy

As a new functional material, shape memory alloy (SMA) can simplify the traditional mechanical structure and design a compact vibration control system. Meanwhile, friction is common in vibration control system, the study of friction is beneficial to the global dynamical analysis. In this paper, the global structure of the dry friction oscillator with SMA is studied by the modified composite cell coordinate system (CCCS) method. By analyzing the influence of excitation amplitude, friction and temperature on global bifurcation, the evolution process of the crisis phenomena with the variation of parameters is illustrated in detail. There exist three kinds of crises including merging crisis, boundary crisis and interior crisis. Moreover, when multiple attractors coexist in the state space, a series of basin boundary metamorphoses, such as smooth-fractal, fractal-Wada and Wada-Wada are also discovered as the temperature and the friction varies. The results show that the modified CCCS method is a powerful tool to reveal the mechanism of crises in the non-smooth system and also demonstrate that the motion of the system can be regulated with the variation of parameters, so as to control the system vibration.

Global analysis of crises in shape memory thin plate system

The unique global properties of shape memory alloy are mainly derived from the martensite phase transition and its inverse, which result from the change of temperature and external load. In this paper, the global characteristics of shape memory alloy thin plate system are analyzed with the temperature and harmonic excitation amplitude as control parameters. Based on the method of Poincare map, the complex crisis phenomenon of the system including the sudden change in number, size and type of attractors can be observed through the global multivalued bifurcation diagram. However, the specific crisis type is not clear, it is necessary to be analyzed from the global viewpoint. By computing the global diagram with the composite cell coordinate system method which constructs a composite cell state space by multistage division of the continuous phase space, the attractors, saddles and basins of attraction of the system can be obtained more accurately. The vivid evolutionary processes of the crisis phenomena of the system are illustrated, and it can be found that the system presents a complex global structure with amplitude and temperature changing. There exist two kinds of crises: one is the boundary crisis resulting from the collision between a chaotic/periodic attractor and a chaotic saddle within the basin boundary, which causes the attractor to vanish, and the other is the merging crisis caused by the collision of two or more attractors with the chaotic saddle within the basin boundary where a new chaotic attractor appears. When multiple attractors coexist in the system, the basin boundary may be smooth or fractal, and for any point at boundary, its small open neighborhood always has a nonempty intersection with three or more basins, which is known as Wada basin boundary. It is difficult to predict the dynamic behavior of the system accurately due to the fractal, the Wada-Wada, Wada-fractal and fractal-Wada basin boundary metamorphoses which can be observed along with the variation of temperature and amplitude through the composite cell coordinate system method, which owns a unique advantage in depicting basin boundary. Furthermore, the Wada property is displayed more clearly by refining specified region. The results of this paper provide a theoretical analysis tool for adjusting the dynamic response of shape memory alloy thin plate system and optimizing the deformation and vibration control of mechanical equipment through controlling temperature and excitation intensity.

Nonlinear dynamics of the quasi-zero-stiffness SD oscillator based upon the local and global bifurcation analyses

In this paper, the complicated nonlinear dynamics of the harmonically forced quasi-zero-stiffness SD (smooth and discontinuous) oscillator is investigated via direct numerical simulations. This oscillator considered that the gravity is composed of a lumped mass connected with a vertical spring of positive stiffness and a pair of horizontally compressed springs providing negative stiffness, which can achieve the quasi-zero stiffness widely used in vibration isolation. The local and global bifurcation analyses are implemented to reveal the complex dynamic phenomena of this system. The double-parameter bifurcation diagrams are constructed to demonstrate the overall topological structures for the distribution of various responses in parameter spaces. Using the Floquet theory and parameter continuation method, the local bifurcation patterns of periodic solutions are obtained. Moreover, the global bifurcation mechanisms for the crises of chaos and metamorphoses of basin boundaries are examined by analysing the attractors and attraction basins, exploring the evolutions of invariant manifolds and constructing the basin cells. Meanwhile, additional nonlinear dynamic phenomena and characteristics closely related to the bifurcations are discussed including the resonant tongues, jump phenomena, amplitude–frequency responses, chaotic seas, transient chaos, chaotic saddles, and also their generation mechanisms are presented.

Metamorphoses of basin boundaries with complex topology in an archetypal oscillator

In a recent paper, we examined a driven archetypal oscillator which is derived from an arch bridge model with a sinusoidally varying central load. We showed that this oscillator can possess complex basins of constrained motions in the space of the initial conditions. We here investigate the mechanisms for the metamorphoses of basin boundary with topological complexity as the control parameter (the magnitude of the excitation) varies. The Wada basin boundary metamorphoses include the creation (or disappearance) of some new (or old) basin sets by a series of saddle-node (or inverse saddle-node) bifurcations in the Poincare map. The changes can also be induced by the collision between the attractors and the stable (or unstable) manifold of the regular saddles. The mechanism for Wada basin boundary metamorphoses is also investigated by the homoclinic tangency and rotary homoclinic tangency. The results perhaps can explain a series of jump phenomena in the mechanical oscillator and give a good guide to safe parameter design according to safe basin.

The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

Although Chua's circuit is one of the most studied nonlinear dynamical systems, its version with negative parameters remains practically untouched. This work reports an interesting and rich dynamic scenery that was hidden in this almost unexplored region. The study is focused on 2D parameter space and presents an analysis of stability based on describing functions. Numerical investigations present a gallery of period-adding cascades and a strong presence of basin boundary metamorphoses. The key to this new scenario is that for negative parameters, Chua's system does not satisfy the Shilnikov condition and it is shown that the homoclinic orbit organizes the parameter space completely different from as known. The obtained experimental results corroborate with the numerical and theoretical investigations.

Wada bifurcations and partially Wada basin boundaries in a two-dimensional cubic map

Abstract The partially Wada basin boundaries are referred to the coexistence of Wada points and non-Wada points in the same basin boundary. We demonstrate two types of Wada bifurcations and analyze the transitions from totally Wada basins to partially Wada basins and from totally Wada basins to totally Wada basins in a two-dimensional cubic map. We describe some numerical experiments giving the evidence of partially Wada basin boundaries. Our results show that the basin cell erosion and the basin cell bifurcation can induce the Wada basin boundary metamorphoses.

Global bifurcation analysis of Rayleigh–Duffing oscillator through the composite cell coordinate system method

In this paper, a new conception of composite cell coordinate system is presented by dividing the continuous state space into the cell state space with different scales. For a dynamical system, attractors, basins of attraction, basin boundaries, saddles, and invariant manifolds can be easily obtained, and any region of the state space can be refined by this method. The global bifurcations, such as crisis and metamorphosis, of the Rayleigh–Duffing oscillator are studied by the composite cell coordinate system method. According to the sudden changes in shapes of the chaotic attractor and the chaotic saddle, we find that three types of crises can all occur, including boundary crisis, interior crisis, and attractor emerging crisis. In addition, the basin boundary metamorphoses, such as fractal-Wada, Wada-Wada, and Wada-fractal, are analyzed through observing the shapes of basin boundaries. These results demonstrate the efficiency and validity of this method in analyzing dynamical systems.

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.

Boundary crisis bifurcation in two parameters

The boundary crisis bifurcation is well known as a mechanism for destroying (or creating) a strange attractor by variation of one parameter: at the moment of the boundary crisis bifurcation the strange attractor touches the boundary of its own basin of attraction. Here we follow this codimension-one bifurcation in two parameters. One might expect that this leads to a differentiable curve in the two-parameter plane. Mathematically, a boundary crisis is effectively a homoclinic or heteroclinic tangency, the locus of which is a well-defined differentiable curve in a two-parameter system. However, instead of a boundary crisis, the transition through this tangency curve may lead to a basin boundary metamorphosis or an interior crisis bifurcation, in which the attractor persists. This phenomenon is again well known: at the point where the type of transition changes, the boundary crisis switches to another branch of homoclinic or heteroclinic tangencies, associated with manifolds of a periodic point with a different period than before. The curve of boundary crisis bifurcations is not differentiable at such points. In this paper, we show that the curve of boundary crisis bifurcations is, in fact, not even well defined as a piecewise-smooth curve. We illustrate that there are infinitely many gaps in much the same way as the one-parameter bifurcation diagram of the attractor contains infinitely many windows where the attractor is periodic and not strange or chaotic. Throughout, we use the Hénon map to illustrate our findings.

Fluctuational transitions across different kinds of fractal basin boundaries

We study fluctuational transitions in discrete and continuous dynamical systems that have two coexisting attractors in phase space, separated by a fractal basin boundary which may be either locally disconnected or locally connected. Theoretical and numerical evidence is given to show that, in each case, the transition occurs via a unique accessible point on the boundary, both in discrete systems and in flows. The complicated structure of the escape paths inside the locally disconnected fractal basin boundary is determined by a hierarchy of homoclinic points. The interrelation between the mechanism of transitions and the hierarchy is illustrated by consideration of fluctuational transitions in dynamical systems demonstrating "fractal-fractal" basin boundary metamorphosis at some value of a control parameter. The most probable escape path from an attractor, which can be either regular or chaotic, is found for each type of boundary using both statistical analysis of fluctuational trajectories and the Hamiltonian theory of fluctuations.

FRACTAL PARAMETERS FOR THE STUDY OF HÉNON-LIKE ATTRACTORS

For invertible, area-contracting planar maps, much work has been done on the study of the crucial role played by certain unstable periodic orbits, distinguished by being "accessible", in understanding some global bifurcations such as metamorphoses of basin boundaries and crisis of attractors. In this paper we concentrate on a one-parameter family of Hénon maps whose attractors present a type of discontinuous change characterized by a sudden replacement of the accessible orbits as the parameter is varied. The change is quantified by a jump in the corresponding accessible rotation number. The rotation rate describes a devil's staircase as a function of the parameter. We estimate the Hausdorff dimensions of the Hénon attractors — via the Mendès France fractal dimension — for the same interval of parameter values. Our purpose is to show the strong connection between these two functions: the variation of the dimension stays minimal for the parameters under the same plateau in the staircase, while it is markedly greater when the parameter moves from one plateau to another.

Mechanism for boundary crises in quasiperiodically forced period-doubling systems

We investigate the mechanism for boundary crises in the quasiperiodically forced logistic map which is a representative model for quasiperiodically forced period-doubling systems. For small quasiperiodic forcing ɛ, a chaotic attractor disappears suddenly via a “standard” boundary crisis when it collides with the smooth unstable torus. However, when passing a threshold value of ɛ, a basin boundary metamorphosis occurs, and then the smooth unstable torus is no longer accessible from the interior of the basin of the attractor. For this case, using the rational approximations to the quasiperiodic forcing, it is shown that a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor is destroyed abruptly through a new type of boundary crisis when it collides with an invariant “ring-shaped” unstable set which has no counterpart in the unforced case.

Why Are Chaotic Attractors Rare in Multistable Systems?

We show that chaotic attractors are rarely found in multistable dissipative systems close to the conservative limit. As we approach this limit, the parameter intervals for the existence of chaotic attractors as well as the volume of their basins of attraction in a bounded region of the state space shrink very rapidly. An important role in the disappearance of these attractors is played by particular points in parameter space, namely, the double crises accompanied by a basin boundary metamorphosis. Scaling relations between successive double crises are presented. Furthermore, along this path of double crises, we obtain scaling laws for the disappearance of chaotic attractors and their basins of attraction.

Catastrophic bifurcation from riddled to fractal basins.

Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In realistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry-breaking on riddling. In particular, we consider smooth deterministic perturbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurcations. In the first case, the chaotic attractor in the invariant subspace is transversely stable so that the basin is riddled. We find that a bifurcation from riddled to fractal basins can occur in the sense that an arbitrarily small amount of symmetry breaking can replace the riddled basin by fractal basins. We call this a catastrophe of riddling. In the second case, where the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a symmetry breaking, no matter how small, can immediately create fractal basins in the vicinity of the original invariant subspace. This is a smooth-fractal basin boundary metamorphosis. We analyze the dynamical mechanisms for both catastrophes of riddling and basin boundary metamorphoses, derive scaling laws to characterize the fractal basins induced by symmetry breaking, and provide numerical confirmations. The main implication of our results is that while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.

Explosions of chaotic sets

Large-scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied. These bifurcations include sudden changes in the size and/or type of the set. An explosion is a bifurcation in which new recurrent points suddenly appear at a non-zero distance from any pre-existing recurrent points. We discuss the following. In a generic one-parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur: (1) a saddle-node bifurcation isolated from other recurrent points, (2) a saddle-node bifurcation embedded in the set of recurrent points, (3) outer homoclinic tangencies, and (4) outer heteroclinic tangencies. (The term “outer tangency” refers to a particular configuration of the stable and unstable manifolds at tangency.) In particular, we examine different types of tangencies of stable and unstable manifolds from orbits of pre-existing invariant sets. This leads to a general theory that unites phenomena such as crises, basin boundary metamorphoses, explosions of chaotic saddles, etc. We illustrate this theory with numerical examples.

Continued Fractions Hierarchy of Rotation Numbers in Planar Dynamics

Global bifurcations such as crises of attractors, explosions of chaotic saddles, and metamorphoses of basin boundaries play a crucial role in understanding the dynamical evolution of physical systems. Global bifurcations in dissipative planar maps are typically caused by collisions of invariant manifolds of periodic orbits, whose dynamical behaviors are described by rotation numbers. We show that the rotation numbers of the periodic orbits created at certain important tangencies are determined by the continued fraction expansion of the rotation number of the orbit involved in the collision.

An Experimental Verification of Basin Metamorphoses in A Nonlinear Mechanical System

This paper describes bifurcations and the basin boundary metamorphoses that give rise to post-fold outcome indeterminacy from a primarily experimental perspective. A gravity-loaded cart-and-track system is constrained to mimic the twin-well, single-degree-of-freedom Duffing oscillator. Of primary interest is the study of how motion, initially contained within a single well, "spills over" into the adjacent well. Although this system is globally bounded, it retains the same generic features of the single-well canonical escape equation. Using time-embedded coordinates, the technique of stochastic interrogation is used to generate the initial condition maps at three different forcing levels corresponding to three different regimes of post-fold outcomes. These three regions are characterized, respectively, by smooth basin boundaries with safe jumps to resonance, fractal basin boundaries with jumps that may or may not restabilize on to the resonant attractor, and eroded basins with unsafe jumps leading to escape from the local well. This experiment successfully replicates much of the subtle global behavior observed in numerical simulations.