Outer Tangency Bifurcations of Chaotic Sets
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Outer Tangency Bifurcations of Chaotic Sets
We present and explain numerical results illustrating the mechanism of a type of discontinuous bifurcation of a chaotic set that occurs in typical dynamical systems. After the bifurcation, the chaotic set acquires new pieces located at a finite distance from its location just before the bifurcation, and these new pieces were not part of a previously existing chaotic set. A scaling law is given describing the creation of unstable periodic orbits following such a bifurcation. We also provide numerical evidence of such a bifurcation for a nonattracting chaotic set of the Hénon map.
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 a nonzero 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, explosion 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 fractions expansion of the rotation number of the orbit involved in the collision.
Double Crises in Two-Parameter Dynamical Systems
A crisis is a sudden discontinuous change in a chaotic attractor as a system parameter is varied. We investigate phenomena observed when two parameters of a dissipative system are varied simultaneously, following a crisis along a curve in the parameter pl ane. Two such curves intersect at a point we call a double crisis vertex. The phenomena we study include the double crisis vertex at which an interior and a boundary crisis coincide, and related forms of double crisis. We show how an experimenter can infe r a crisis from observations of other related crises at a vertex.
Crisis and Enhancement of Chaotic Scattering
Chaotic scattering is characterized by the existence of nonattracting chaotic invariant sets in phase space. There can be several chaotic invariant sets coexisting in phase space when a system parameter value is below some critical value. As the parameter changes through the critical value, stable and unstable foliations of these chaotic invariant sets, which are fractal sets, can become tangent and then cross each other. The first tangency, which provides the linking between chaotic invariant sets, is a crisis in chaotic scattering. Above the crisis, there is an infinite number of such tangencies which keep occuring until the last tangency, above which the stable and unstable foliations cross transversely. As a consequence of this the fractal dimension o f the set of singularities in the scattering function increases in the parameter range determined by the first and the last tangencies. This leads to a proliferation of singularities in the scattering function and consequently, to an enhancement of chaoti c scattering. The phenomenon is investigated by using both simple one-dimensional models and a two-dimensional physical scattering system.
Vertices in Parameter Space: Double Crises which Destroy Chaotic Attractors
We report a new phenomenon observed along a crisis locus when two control parameters of physical models are varied simultaneously: the existence of one or several vertices. The occurrence of a vertex (loss of differentiability) on a c risis locus implies the existence of simultaneous sudden changes in the structure of both the chaotic attractor and of its basin boundary. Vertices correspond to degenerate tangencies between manifolds of the unstable periodic orbits acc essible from the basin of the chaotic attractor. Physically, small parameter perturbations (noise) about such verticies induce drastic changes in the dynamics.
Experimental Observation of Crises Induced Intermittency and its Critical Exponent
Critical behavior associated with intermittent temporal bursting accompanying the sudden widening of a chaotic attractor was observed and investigated experimentally in a gravitationally buckled, parametrically driven, magnetoelastic ribbon. As the driving frequency, f, was decreased through the critical value, fc, we observed that the mean time between bursts scaled as the absolute value of fc - f to a power of -g.
Critical Exponents for Crisis Induced Intermittency
We consider three types of changes that attractors can undergo as a system parameter is varied. The first type leads to the sudden destruction of a chaotic attractor. The second type leads to the sudden widening of a chaotic attractor. In the third type of change, which applies for many systems with symmetries, two ( or more) chaotic attractors merge to form a single chaotic attractor and the merged attractor can-be larger in phase-space extent than the union of the attractors before the change. All three of these types of changes are termed crises and are accompanied by a characteristic temporal behavior of orbits after the crisis. For the case where the chaotic attractor is destroyed, this characteristic behavior is the existence of chaotic transients. For the case where the chaotic attractor suddenly widens, the characteristic behavior is an intermittent bursting out of the phase-space region within which the attractor was confined before the crisis. For the case where the attractors suddenly merge, the characteristic behavior is an intermittent switching between behaviors characteristic of the attractors before merging. In all cases a time scale T can be defined which quantifies the observed post-crisis behavior: for attractor destruction, T is the average chaotic transient lifetime; for intermittent bursting, it is the mean time between bursts; for intermittent switching it is the mean time between switches. The purpose of this paper is to examine the dependence of T on a system parameter (call it p) as this parameter passes through its crisis value p = pc. Our main result is that for an important class of systems the dependence of T on p is T is proportional to p-pc raised to a power g for p close to pc , and we develop a quantitative theory for the determination of the critical exponent g. Illustrative numerical examples are given. In addition, applications to experimental situation, as well as generalizations to higher-dimensional cases, are discussed. Since the case of attractor destruction followed by chaotic transients has previously been illustrated with examples [C. Grebogi, E. Ott and J. A.Yorke, Phys. Rev. Lett. 57, 1284 (1986)], the numerical examples reported in this paper will be for crisis-induced intermittency (i.e., intermittent bursting and switching).
Crises, Sudden Changes in Chaotic Attractors and Chaotic Transients
The occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated. It is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attracto r. We call such collisions crises. Phenomena associated with crises include sudden changes in the size of chaotic attractors, sudden appearances of chaotic attractors (a possible route to chaos), and sudden destructions of chaotic attractors and th eir basins. This paper presents examples illustrating statistical behavior (whose type depends on the type of crisis) occurs. In particular the phenomenon of chaotic transients is investigated. The examples discussed illustrate crises in progressively hig her dimension and include the one-dimensional quadratic map, the (two-dimensional) Hénon map, systems of ordinary differential equations is proposed which is possible only in invertible maps or flows of dimension at least three or four, respectivel y. Based on the examples presented the following conjecture is proposed: almost all sudden changes in the size of chaotic attractors and almost all sudden destructions or creations of chaotic attractors and their basins are due to crises.
Chaotic Attractors in Crisis
The occurrence of sudden qualitative changes of chaotic (or turbulent ) dynamics is discussed and illustrated within the context of the one-dimensional quadratic map. For this case, the chaotic region can suddenly widen or disappear, and the cause and properties of these phenomena are investigated.
Effect of Noise on Nonhyperbolic Chaotic Attractors
We consider the effect of small noise of maximum amplitude epsilon on a chaotic system whose noiseless trajectories limit on a fractal strange attractor. For the case of nonhyperbolic attractors of two-dimensional maps the effect of noise can be made much stronger than for hyperbolic attractors. In particular, the maximum over all noisy orbit point of the distance between the noisy orbit and the noiseless nonhyperbolic attractor scales like epsilon1/D (D is the information dimension of the attractor), rather than like epsilon (the hyperbolic case). We also find a phase transition in the scaling of the time averaged moments of the deviations of a noisy orbit from the noiseless attractor.
Predicting Chaos Most of the Time from Embeddings with Self-Intersection
Embedding techniques for predicting chaotic time series from experimental data may fail if the reconstructed attractor self-intersects, and such intersections often occur unless the embedding dimension exceeds twice the attractor's box counting dimension. Here we consider embedding with self-intersection. When the dimension M of the measurement space exceeds the information dimension D1 of the attractor, reliable prediction is found to be still possible from most orbit points. In particular, the fraction of state space measure from which prediction fails typically scales as epsilon M-D1 for small epsilon where epsilon is the diameter of the neighborhood current state used for prediction.
Correlation Dimension for Iterated Function Systems
The correlation dimension of an attractor is a fundamental dynamical invariant that can be computed from a time series. We show that the correlation dimension of the attractor of a class of iterated function systems in RN is typically uniquely determined by the contraction rates of the maps which make up the system. When the contraction rates are uniform in each direction, our results imply that for a corresponding class of deterministic systems the information dimension of the attractor is typically equal to its Lyapunov dimension, as conjectured by Kaplan and Yorke.
Fractal Properties of Critical Invariant Curves
We examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers, through both the thermodynamic formalism and numerical computation. The maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed. Our results indicate that the dimension is universal for a given type of singularity and rotation number, and that among all rotation numbers, the golden mean produces the largest dimension.
Fractal Dimensions of Chaotic Saddles of Dynamical Systems
A formula, applicable to invertible maps of arbitrary dimensionality, is derived for the information dimensions of the natural measures of a nonattracting chaotic set and of its stable and unstable manifolds. The result gives these dimensions in terms of the Lyapunov exponents and the decay time of the associated chaotic transient. As an example, the formula is applied to the physically interesting situation of filtering of data from chaotic systems.
Fractal dimension fluctuations for snapshot attractors of random maps
We consider the determination of the information dimension of a fractal snapshot attractor (i.e., the pattern formed by a cloud of orbits at fixed time) of a random map. It is found that box-counting estimates of the dimension fluctuate from r ealization to realization of the random process. These fluctuations about the true dimension value are a result of the unavoidable presence of a finite smallest box size epsilon* used in the dimension estimation. The main result is that the fluctuations a re well-described by a Gaussian probability distribution function whose width is proportional to (log 1/epsilon*)^(-1/2). Averaging dimension estimates over many realizations (or over time for a single realization) thus yields a means of obtaining a great ly improved estimate of the true dimension value.
When Cantor Sets Intersect Thickly
The thickness of a Cantor set on the real line is a measurement of its size. Thickness conditions have been used to guarantee that the intersection of two Cantor sets is nonempty. We present sharp conditions on the thicknesses of two Cantor sets which imply that their intersection contains a Cantor set of positive thickness.
Estimating Correlation Dimension from a Chaotic Time Series: When Does Plateau Onset Occur?
Suppose that a dynamical system has a chaotic attractor A with a correlation dimension D2. A common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m - dimensional space using the delay-coordinate technique. The estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau (on which the dimension estimate if relatively constant) for a range of large enough m values. The plateaued dimension value is then assumed to be an estimate of D2 for the attractor in the original full phase space. In this paper we first present rigorous results which state that, for a long enough data string with low enough noise, the plateau onset occurs at m = Ceil (D2), where Ceil (D2), standing for ceiling of D2, is the smallest integer greater than or equal to D2. We then show numerical examples illustrating the theoretical prediction. In addition, we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m = Ceil (D2).
Plateau Onset for Correlation Dimensions: When does it Occur?
Chaotic experimental systems are often investigated using delay coordinates. Estimated values of the correlation dimension in delay coordinate space typically increase with the number of delays and eventually reach a plateau (on which the dime nsion estimate is relatively constant) whose value is commonly taken as an estimate of the correlation dimension D2 of the underlying chaotic attractor. We report a rigorous result which implies that, for long enough data set s, the plateau begins when the number of delay coordinates first exceed D2. Numerical experiments are presented. WE also discuss how lack of sufficient data can produce results that seem to be inconsistent with the theoretica l prediction.
The Equality of Fractal Dimension and Uncertainty Dimension for Certain Dynamical
Systems
MGOY [C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Phys. Lett. 110A (1985)] introduced the uncertainty dimension as a quantitative measure for final state sensitivity in a system. In MGOY it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.
Lyapunov Partition Functions for the Dimensions of Chaotic Sets
Multifractal dimension spectra for the stable and unstable manifolds of invariant chaotic sets are studied for the case of invertible two-dimensional maps. A dynamical partition-function formalism giving these dimensions in terms of local Lyapunov numbers is obtained. The relationship of the Lyapunov partition functions for stable and unstable manifolds to previous work is discussed. Numerical experiments demonstrate that dimension algorithms based on the Lyapunov partition function are often very efficient. Examples supporting the validity of the approach for hyperbolic chaotic sets and for nonhyperbolic sets below the phase transition (q < qT) are presented.
Theory of First Order Phase Transitions for Chaotic Attractors of Nonlinear Dynamical Systems
A theory is presented for first order phase transitions of multifractal chaotic attractors of nonhyperbolic two-dimensional maps. (These phase transitions manifest themselves as a discontinuity in the derivative with respect to q (analogous to temperature) of the fractal dimension q-spectrum, Dq (analogous to free energy).) A complete picture of the behavior associated with the phase transition is obtained.
Roundoff-Induced Periodicity and the Correlation Dimension of Chaotic Attractors
Due to roundoff, digital computer simulations of orbits on chaotic attractors will always eventually become periodic. The expected period, probability distribution of periods, and expected number of periodic orbits are investigated for the case of fractal chaotic attractors. The expected period scales with roundoff epsilon as epsilon-d/2, where d is the correlation dimension of the chaotic attractor.
Unstable Periodic Orbits and the Dimensions of Multifractal Chaotic Attractors
The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily fine-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and chaotic repellers are considered.
Unstable Periodic Orbits and the Dimension of Chaotic Attractors
A formulation giving the q dimension Dq of a chaotic attractor in terms of the eigenvalues of unstable periodic orbits is presented and discussed.
Exterior Dimension of Fat Fractals
Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which we call the exterior dimension. In addition, it is shown that the exterior dimension is related to the uncertainty exponent previously used in studies of fractal basin boundaries, and it is show how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.
A Scaling law: How an Attractor's Volume Depends on Noise Level
We investigate the meaning of the dimension of strange attractor for systems with noise. More specifically, we investigate the effect of adding noise of magnitude g to a deterministic system with D degrees of freedom. If the attractor has dimension d and d < D, then its volume is zero. The addition of noise may be an important physical probe for experimental situations, useful for determining how much of the observed phenomena in a system is due to noise already present. When the noise is added the attractor Ag has positive volume. We conjecture that the generalized volume of Ag is proportional to gD - d for g near 0 and show this relationship is valid in several cases. For chaotic attractors there are a variety of ways of defining d and the generalized volume definition must be chosen accordingly.
The Lyapunov Dimension of a Nowhere Differentiable Attracting Torus
The fractal dimension of an attracting torus T k in R X T k is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have several Lyapunov numbers greater than 1 and several less than 1.
Fat Baker's Transformations
We investigate a variant of the baker transformation in which the mapping is onto but is not one-to-one. The Bowen-Ruelle measure for this map is investigated.
Is the Dimension of Chaotic Attractors Invariant Under Coordinate Changes?
Several different dimension-like quantities, which have been suggested as being relevant to the study of chaotic attractors, are examined. In particular, we discuss whether these quantities are invariant under changes of variables that are differentiable except as a finite number of points. It is found that some are and some are not. It is suggested that the word dimension be reversed only for those quantities that have this invariance property.
The Lyapunov Dimension of Strange Attractors
Many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange. In this paper the question of estimating the dimension of the attractor is addressed. While more general conjectures are made here, particular attention is paid to the idea that if the Jacobian determinant of a map is greater than one and a ball is mapped into itself, then generically, the attractor will have positive two-dimensional measure, and most of this paper is devoted to presenting cases with such Jacobians for which the attractors are proved to have non-empty interior.
Spurious Lyapunov Exponents in Attractor Reconstruction
Lyapunov exponents, perhaps the most informative invariants of a complicated dynamical process, are also among the most difficult to determine from experimental data. In particular, when using embedding theory to build chaotic attractors in a reconstruction space, extra spurious Lyapunov exponents arise that are not Lyapunov exponents of the original system. The origin of these spurious exponents is discussed, and formulas for their determination in the low noise limit are given.
Are the Dimensions of a Set and Its Image Equal Under Typical Smooth Functions?
We examine the question whether the dimension D of a set or probability measure is the same as the dimension of its image under s typical smooth function, if the phase space is at least D-dimensional. If m is a Borel probability measure of bounded support in Rn with correlation dimension D, and if k = D, then under almost every continuously differentiable function (almost every in the sense of prevalence) from Rn to Rm, the correlation dimension of the image of m is also D. If m is the invariant measure of a dynamical system, the same is true for almost every delay coordinate map, under weak conditions on periodic orbits. That is, if k = D, the k time delays are sufficient to find the correlation dimension using a typical measurement function. Further, it is shown that finite impulse response (FIR) filters do not change the correlation dimension. Analogous theorems hold for Hausdorff, pointwise, and information dimension. We show by example that the conclusion fails for box-counting dimension.
How Many Delay Coordinated Do You Need?
Theorems on the use of delay coordinates for analyzing experimental data are discussed. To reconstruct a one-to-one correspondence with the state-space attractor, m delay coordinates are sufficient, where m 2D0 (here D0 denotes the box-counting dimension). For calculating the correlation dimension D2, m D2 delays are sufficient. These results remain true under finite impulse (FIR) filters.
Prevalence: A Translation-Invariant "Almost Every" on Infinite-Dimensional Soaces
We present a measure-theoretic condition for a property to hold almost everywhere on an infinite-dimensional vector space, with particular emphasis on function spaces such as C k and Lp. Like the concept of Lebesgue almost every on finite-dimensional spaces, our notion of prevalence is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in terms of the class of all probability measures with compact support. Prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space. We give several examples of properties which hold almost everywhere in the sense of prevalence. For instance, we prove that almost ever C 1 map on Rn has the property that all of its periodic orbits are hyperbolic.
Embedology
Mathematical formulations of the embedding methods commonly used for the reconstruction of attractors from data series are discussed. Embedding theorems, based on previous work by H. Whitney and F. Takens, are established for compact subsets A of Euclidean space Rk. If n is an integer larger than twice the box-counting dimension of A, then almost every map from Rk to Rn, in the sense of prevalence, is one-to-one on A, and moreover is an embedding on smooth manifolds contained within A. If A is a chaotic attractor of a typical dynamical system, then the same is true for almost every delay-coordinate map from Rk to Rn. These results are extended in two other directions. Similar results are proved in the more general case of reconstructions which use moving averages of delay coordinates. Second, information is given on the self-intersection set that exists when n is less than or equal to twice the box-counting dimension of A.
Using Dynamic Embedding Method to Analyze Experimental Data
The time delay embedding method provides a powerful tool for the analysis of experimental data. We show how recent improvements allow experimentalists to use many of the same techniques that have been essential to the analysis of nonlinear systems of ordinary differential equations and difference equations.
Quasi-two-dimensional fast kinematic dynamo instabilities of chaotic fluid flows
This paper tests previous heuristically derived general theoretical results for the fast kinematic dynamo instability of a smooth, chaotic flow by comparison of the theoretical results with numerical computations on a particular class of model flows. The class of chaotic flows studied allows very efficient high resolution computation. It is shown that an initial spatially uniform magnetic field undergoes two phases of growth, one before and one after the diffusion scale has been reached. Fast dynamo action is obtained for large magnetic Reynolds number R_m. The initial exponential growth rate of moments of the magnetic field, the long time dynamo growth rate, and multifractal dimension spectra of the magnetic fields are calculated from theory using the numerically determined finite time Lyapunov exponent probability distribution of the flow and the cancellation exponent. All these results are numerically tested by generating a quasi-two-dimensional dynamo at magnetic Reynolds number R_m of order up to 10^5.
Fractal Dimensions of Fast Dynamo Magnetic Fields
Recent studies of the fast kinematic dynamo find that, without diffusion, the magnetic flux tends to concentrate on a fractal set and generally exhibits an arbitrarily fine-scaled alternation in the orientation of the magnetic field. The prese nce of small magnetic diffusivity in conjunction with this arbitrarily fine-scaled alternation in the magnetic field orientation radically changes the small scale spatial structure of the field. Motivated by this consideration, we study the multifractal d imension spectra associated with fast dynamo magnetic fields both with and without diffusion. These spectra are shown to be different in the two cases. In addition, using a partition function technique, we obtain results giving the dimension spectra in te rms of the distribution of finite time Lyapunov exponents. Numerical experiments illustrating and test these results are presented.
Sign-Singular Measures: Fast Magnetic Dynamos and High Reynolds Number Fluid Turbulence
It is shown that signed measures (i.e., measures that take on both positive and negative values) may exhibit an extreme form of singularity in which oscillations in sign occur everywhere on arbitrarily fine scale. A cancellation exponent is introduced to characterize such measures quantitatively, and examples of significant physical situations which display this striking type of singular behavior are discussed.
Chaotic Flows and Magnetic Dynamos
Examples of time-dependent kinematic magnetic dynamos arising from chaotic flows in the zero resistivity limite are presented. These examples suggest that typically (1) the flux concentrates on a fractal, and (2) the magnetic field exhibits ar bitrarily fine-scaled oscillations between parallel and antiparallel directions. Results quantitatively relating these effects to the dynamo growth rate are presented.
Wada Dye Boundaries in Open Hydrohynamical Flows
Dyes of different colors advected by two-dimensional flows which are asymptotically simple can form a fractal boundary that coincides with the unstable manifold of a chaotic saddle. We show that such dye boundaries can have the Wada property: every boundary point of a given color on this fractal set is on the boundary of at least two other colors. The condition for this is the nonempty intersection of the stable manifold of the saddle with at least three differently colored domains in the asymptotic inflow region.
The Structure of Basins of Attraction and Their Trapping Regions
In dynamical systems examples are common in which two or more attractors coexist, and in such cases, the basin boundary is nonempty. When there are three basins of attraction, is it possible that every boundary point of one basin is on the boundary of the two remaining basins? Is it possible that all three boundaries of these basins coincide? When this last situation occurs the boundaries have a complicated structure. This phenomenon does occur naturally in simply dynamical systems. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. We introduce the basic notion of a basin cell. A basin cell is a trapping region generated by some well chosen periodic orbit and determines the structure of the corresponding basin. This new notion will play a fundamental role in our main results. We consider diffeomorphisms of a two-dimensional smooth manifold M without boundary, which has at least three basins. A point x in M is a Wada point if every open neighborhood of x has a nonempty intersection with at least three different basins. We call a basin B a Wada basin if every x in the boundary of the closure of B is a Wada point. Assuming B is the basin of a basin cell (generated by a periodic orbits P), we show the B is a Wada basin if the unstable manifold of P intersects at least three basins. This result implies conditions for basins B1, B2,..., BN (N=3) to all have exactly the same boundary.
Basins of Attraction
Many remarkable properties related to chaos have been found in the dynamics of nonlinear physical systems. These properties are often seen in detailed computer studies, but it is almost always impossible to establish these properties rigorously for specific physical systems. This article presents some strange properties about basins of attraction. In particular, a basin of attraction is a Wada basin if every point on the common boundary of that basin and another basin is also on the boundary of a third basin. The occurrence of this strange property can be established precisely because of the concept of a basin cell.
Wada Basin Boundaries and Basin Cells
In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty. We consider a two-dimensional diffeomorphism F (that is, F is an invertible map and both F and its inverse are differentiable with continuous derivatives), which has at least three basins. Fractal basin boundaries contain infinitely many periodic points. Generally, only finitely many of these periodic points are outermost on the basin boundary, that is, accessible from a basin. For many systems, all accessible points lie on stable manifolds of periodic points. A point x on the basic boundary is a Wada point if every open neighborhood of x has a nonempty intersection with at least three different basins. We call the boundary of a basin a Wada basin boundary if all its points are Wada points. Our main goal is to have definitions and hypotheses for Wada basin boundaries that can be verified by computer. The basic notion basin cell will play a fundamental role in our results for numerical verifications. Assuming each accessible point on the boundary of a basin B is on the stable manifold of some periodic orbit, we show that the boundary of the closure of B is a Wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins. In addition, we find condition for basins B1, B2,..., BN (N 2) under which all Bi have the same boundary. Our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a Wada basin boundary. Our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of Wada basin boundaries.
Saddle-Node Bifurcations on Fractal Basin Boundaries
We demonstrate and analyze a bifurcation producing a type of fractal basin boundary which has the strange property that any point which is on the boundary of that basi is also simultaneously on the boundary of at least two other basins. We giv e rigorous general criteria guaranteeing this phenonmenon, present illustrative numerical examples, and discuss the practical significance of the results.
Accessible Saddles on Fractal Basin Boundaries
For a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected, and simply-connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When i t is not, it can contain periodic orbits of infinitely many different periods. Certain points on the basin boundary are distinguished by bein accessible (by a path) from the interior of the basin. For an orientation-preserving homeomorphism, the accessibl e boundary points have a well-defined rotation number. Under some genericity assumptions, we prove that this rotation number is rational if and only iff there are accessible periodic orbits. In particular, if the rotation number is the reduced fraction p/q and if the periodic orbits of periods q and smaller are isolated, then every accessible periodic orbit has minimum period q. In addition, if the periodic orbits are hyperbolic, then every accessible point is on the stable manifold o f an accessible periodic point.
Basins of Wada
We describe situations in which there are several regions (more than two) with the Wada property, namely that each point that is on the boundary of one region in on the boundary of all. We argue that such situations arise even in studies of th e forced damped pendulum, where it is possible to have three attractor regions coexisting, and the three basins of attraction have the Wada property.
Metamorphoses: Sudden Jumps in Basin Boundaries
In some invertible maps of the plane that depend on a parameter, boundaries of basins of attraction are extremely sensitive to small changes in the parameter. A basin boundary can jump suddenly, and, as it does, change from being smooth to fractal. Such changes are call basin boundary metamorphoses. We prove (under certain non-degeneracy assumptions) that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency.
Scaling of Fractal Basin Boundaries Near Intermittency Transitions to Chaos
It is the purpose of this paper to point out that the creation of fractal basin boundaries is a characteristic feature accompanying the intermittency transition to chaos. (Here intermittency transition is used in the sense of Pomeau and Manneville [Commun. Math. Phys. 74, 189 (1980)]; viz., a chaotic attractor is created as a periodic orbit becomes unstable.) In particular, we are here concerned with type-I and type-III intermittencies. We examine the scaling of the dimension of basin boundaries near these intermittency transition. We find, from numerical experiments, that near the transition the dimension scales with a system parameter p according to the power law D is asymptotically like d0-k [p-pI], where d0 is the dimension at the intermittency transition parameter value p = pI and k is a scaling constant. Furthermore, for type-I intermittency d0 < D, while for type-III intermittency d0 = D, where D is the dimension of the space. Heuristic analytic arguments supporting the above are presented.
Fractal Boundaries for Exit in Hamiltonian Dynamics
Fractal boundaries can occur for certain situations involving chaotic Hamiltonian systems. In particular, situations are considered in which an orbit can exit from the system in one of several different ways, and the question is asked which of these ways applies for a given initial condition. As an illustration, specific examples are considered for which there are two possible ways in which a particle can exit from the system. We examine the space of initial conditions to see which of the two exit possibilities applies for each initial condition. It is found that the regions of initial-condition state space corresponding to the two exit modes are separated by a boundary that has both fractal and smooth (nonfractal) regions, for one example, an d by a fractal boundary for the other example. Furthermore, it is found for the example where the boundary has fractal and smooth regions that these regions are intertwined on arbitrarily fine scale. The existence of fractal boundaries is conjectured to b e a typical property of chaotic Hamiltonian dynamics with multiple exit modes. Two situations in space physics where our results may be relevant are discussed.
Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics
Recently research has shown that many simple nonlinear deterministic systems can behave in an apparently unpredictable and chaotic manner. This realization has broad implications for many fields of science. Basic developments in the field of chaotic dynamics of dissipative systems are reviewed in this article. Topics covered include strange attractors, how chaos comes about with variation of a system parameter, universality, fractal basin boundaries and their effect on predictability, and applications to physical systems.
Basin Boundary Metamorphoses: Changes in Accessible Boundary Orbits
Basin boundaries sometimes undergo sudden metamorphoses. These metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal, or else can cause a fractal basin boundary to suddenly jump in size and change its char acter (although remaining fractal). For an invertible map in the plane, there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal. Nonetheless, we have found that typically only one of them can be reached or "accessed" directly from a given basin. The other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary. The boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary's ac cessible orbit.
Multi-Dimensioned Intertwined Basin Boundaries and the Kicked Double Rotor
Using numerical computations on a map which describes the time evolution of a particular mechanical system in a four-dimensional phase space (The kicked double rotor), we have found that the boundaries separating basins of attraction can have different properties in different regions and that these different regions can be intertwined on arbitrarily fine scale. In particular, for the double rotor map, if one chooses a restricted region of the phase space and examines the basin boundary in that region, then either one observes that the boundary is a smooth three-dimensional surface or one observes that the boundary is fractal with dimension d – 3.9, and which of these two possibilities applies depends on the particular phase space region chosen for examination. Furthermore, for any region (no matter how small) for which d – 3.9, one can choose subregions within it for which d = 3. (Hence d – 3.9 region and d = 3 region are intertwined on arbitrarily fine scale.) Other examples will also be presented and analyzed to show how this situation can arise. These include one-dimensional map cases, a map of the plane and the Lorenz equations. In one of our one-dimensional map cases the boundary will be fractal everywhere, but the dimension can take on either of two different values both of which lie between 0 and 1. These examples lead us to conjecture that basin boundaries typically can have at most a finite number of possible dimension values. More specifically, let these values be denoted d1, d2,...,dN. Choose a volume region of phase space whose interior contains some part of the basin boundary and evaluate the dimension of the boundary in that region. Then our conjecture is that for all typical volume choices, the evaluated dimension within the chosen volume will be one of the values d1, d2,...,dN. For example, in our double rotor map it appears that N = 2, and d1 = 3.0 and d2 = 3.9.
Metamorphoses of Basin Boundaries in Nonlinear Dynamical Systems
A basin boundary can undergo sudden changes in its character as a system parameter passes through certain critical values. In particular, basin boundaries can suddenly jump in position and can change from being smooth to being fractal. We describe these changes (metamorphoses) and find that they involve certain special unstable orbits on the basin boundary which are accessible from inside one of the basins. The forced damped pendulum (Josephson junction) is used to illustrate these phenomena.
Structure and Crises of Fractal Basin Boundaries
We discuss the structure of fractal basin boundaries in typical nonanalytic maps of the plane and describe a new type of crisis phenomenon.
Fractal Basin Boundaries
Basin boundaries of dynamical systems can either smooth or fractal. This paper investigates fractal basin boundaries. One practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a sys tem eventually goes. The structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected. Examples and discussion of both types of structures are given, and it appears that fractal basin boundaries should be common in typical dynamical systems. Lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed.
k Spectrum of Passive Scalars in Lagrangian Chaotic Fluid Flows
An eikonal-type description for the evolution of k spectra of passive scalars conveted in a Lagrangian chaotic fluid flow is shown to accurately reproduce results from orders of magnitude more time consuming computations based on the fu ll passive scalar partial differential equation. Furthermore, the validity of the reduced description, combined with concepts from chaotic dynamics, allows new theoretical results on passive scalar k spectra to be obtained. Illustrative application s are presented to long-time passive scalar decay, and to Batchelor's law k spectrum and its diffusive cutoff.
Fractal Boundaries in Open Hydrodynamical Flows: Signatures of Chaotic Saddles
We introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently. It is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors. The fractal parts of the boundaries found in the two experiments contain either the stable or unstable manifold of this chaotic set. We point out that these boundaries coin cide with streak lines passing through appropriately chosen points. As an illustrative numerical experiment, we consider a model of the von Karman vortex street, a time periodic two-dimensional flow of a viscous fluid around a cylinder.
Surface Flows, Random Maps, and Fractal Dimensions
Measurements of the local dynamics on the surface of a fluid undergoing complicated motion allow prediction of the measured fractal dimension of an aggregate of passive, floating tracers. This realization of a strange attractor in physical space is a rare instance where there is a firm quantitative connection between the dimension of an experimentally observed fractal spatial pattern and the process producing it. The results also show that the fractal dimension of a tracer distribution is a potential diagnostic for the dynamics of the underlying flow. The analysis of experimental results is performed in terms of random mappings, indicating that this mathematical abstraction provides a potentially useful conceptual framework to bring some spatiotemporal problems into the domain of low-dimensional dynamics.
Particles Floating on a Moving Fluid: A Dynamically Comprehensible Physical Fractal
Measurements of the local dynamics on the surface of a fluid undergoing complicated motion allow prediction of the measured fractal dimension of an aggregate of passive, floating tracers. This realization of a strange attractor in physical space is a rare instance where there is a firm quantitative connection between the dimension of an experimentally observed fractal spatial pattern and the process producing it.
Fractal Distribution of Floaters on a Fluid Surface and the Transition to Chaos for Random Maps
The long-time spatial distribution of particles floating on the surface of a confined fluid whose flow velocity has complicated time dependence is considered. It is shown that this distribution can be either a fractal or else can clump at several (or one) discrete points. The transition from the latter type of distribution to the former occurs when the Lyapunov exponent charactering the particle motion passes through zero form negative values to positive values. The characteristic features of this type of transition are investigated using random maps. It is shown that near the transition there are extremely intermitten temporal fluctuations in the particle cloud, and their scaling with a parameter is elucidated.
Lorenz-like Chaos in a Partial Differential Equation for a Heated Fluid Loop
A set of partial differential equations are developed describing fluid flow and temperature variation in a thermosyphon with particularly simple external heating. Several exact mathematical results indicate that a Bessel-Fourier expansion should converge rapidly to a solution. Numerical solutions for the time-dependent coefficients of that expansion exhibit a transition to chaos like that shown by the Lorenz equations over a wide range of fluid material parameters.
Preturbulence: A Regime Observed in a Fluid Flow Model of Lorenz
This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value which is the first value for which the system possesses a homoclinic orbit.
Metastable Chaos: The Transition to Sustained Chaotic Behavior in the Lorenz Model
The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbers r somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence upon r is studied both numerically and (very close to the critical r) analytically.
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