Highly
Optimized Tolerance in Epidemic Models Incorporating Local Optimization
and Regrowth
All stable processes we shall predict. All unstable processes we shall control - John Von Neumann.
Here, you will find abstracts for some of the paper listed in the papers section. Since this file has become a little bit unwieldly, I have decided to split it into four files for easier manageability. Hopefully this will all be transparent to you!
The papers are further subdivided into the following categories for your convenience, and if you wish, you can go back to the Papers Section
Highly
Optimized Tolerance in Epidemic Models Incorporating Local Optimization
and Regrowth
In the context of a coupled map model of population dynamics, which includes the rapid spread of fatal epidemics, we investigate the consequences of two new features in Highly Optimized Tolerance (HOT). HOT is a mechanism which describes how complexity arises in systems which are optimized for robust performance in the presence of a harsh external environment. Specifically, we (1) contrast global and local optimization criteria and (2) investigate the effects of time dependent regrowth. We find that both local and global optimization lead to HOT states, which may differ in their specific layouts, but share many qualitative features. Time dependent regrowth leads to HOT states which deviate from the optimal configurations in the corresponding static models in order to protect the system from slow (or impossible) regrowth which follows the largest losses and extinctions. While the associated map can exhibit complex, chaotic solutions, HOT states are always in relatively simple dynamical regimes.
Modeling Continuous Mixing of Granular Solids in a Rotating Drum
We consider the avalanche mixing of a monodisperse collection of granular solids in a slowly rotating drum. Although not yet well understood, this process has been studied experimentally for the case where the drum rotates slowly enough that each avalanche ceases completely before a new one begins. We develop a mathematical model for the mixing in both the discrete avalanche case and in the more useful case where the drum is rated quickly enough to induce a continuous avalanche in the material but slowly enough to avoid significant inertial effects. This continuous model in turn provides a more plausible model of the discrete avalanche case. Although avalanches are inherently a nonlinear phenomenon, the mathematical model developed here reduces to a linear integral equation. The asymptotic behavior of the solution for an arbitrary initial distribution is consistent with those obtained experimentally.
Border-Collision Bifurcations in the Buck Converter
Interesting bifurcation phenomena are observed for the current feedback-controlled buck converter. We demonstrate that most of these bifurcations can be categorized as border-collision bifurcation. A method of predicting the local bifurcation structure through the construction of a normal form is applied. This method applies to many power electronic circuits as well as other piecewise smooth systems.
Modeling Fractal Entrainment Sets of Tracers Advected by Chaotic Temporally Irregular Fluid Flows Using Random Maps
We model a two-dimensional open fluid flow that has temporally irregular time dependence by a random map xn + 1 = Mn (xn), where on each iterate n, the map Mn is chosen from an ensemble. We show that a tracer distribution advected through a chaotic region can be entrained on a set that becomes fractal as time increases. Theoretical and numerical results on the multifractal dimension spectrum are presented.
Indecomposable Continua and the Characterization of Strange Sets in Nonlinear Dynamics
We discuss a topological property which we believe provides a useful conceptual characterization of a variety of strange sets occurring in nonlinear dynamics (e.g., strange attractors, fractal basin boundaries, and stable and unstable manifolds of chaotic saddles). Sets with this topological property are known as indecomposable continua. As an example, we give detailed results for the case of an indecomposable continuum that arises from the entrainment of dye advected by a fluid flowing past a cylinder. We show for this case that the indecomposable continuum persists in the presence of small noise.
Topology of Stirred Fluids
There are simple idealized mathematical models representing the stirring of fluids. The models we consider involve two fluids entering a chamber, with the overflow leaving it. The stirring created a Cantor-like, but connected, boundary between the fluids that is best described point-set topologically. We prove that in many cases the boundary between the fluids is an indecomposable continuum.
Indecomposable Continua in Dynamical Systems with Noise: Fluid Flow Past an Array of Cylinders
Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x0 and an initial time t0. We discuss when the set of initial points at a time t0 whose trajectory (x(t), y(t)) is semibounded (i.e., x(t) x0 for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise.
Bifurcation Rigidity
Bifurcation diagrams of periodic windows of scalar maps are often found to be not only topologically equivalent, but in fact to be related by a nearly linear change of parameter coordinates. This effect has been observed numerically for one-parameter families of maps, and we offer an analytical explanation for this phenomenon. We further present numerical evidence of the same phenomenon for two-parameter families, and give a mathematical explanation like that for the one-parameter case.
Robust Chaos
Practical applications of chaos require the chaotic orbit to be robust, defined by the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space. We show that robust chaos can occur in piecewise smooth systems and obtain the conditions of its occurrence. We illustrate this phenomenon with a practical example from electrical engineering.
From High Dimension Chaos to Stable Periodic Orbits: The Structure of Parameter Space
Regions in the parameter space of chaotic systems that correspond to stable behavior are often referred to as windows. In this Letter, we elucidate the occurrence of such regions in higher dimensional chaotic systems. We describe the fundamental structure of these windows, and also indicate under what circumstances one can expect to find them. These results are applicable to systems that exhibit several positive Lyapunov exponents, and are of importance to both the theoretical and the experimental understanding of dynamical systems.
Persistent Homoclinic Tangencies in the Henon Family
Homoclinic tangencies in the Henon family fa (x, y) = (a - x2 + by, x) for the parameter values b = 0.3 and a in [1.270, 1.420] are investigated. Our main observation is that there exist three intervals comprising 93 percent of the values of the parameter 8 such that for a dense set of parameter values in these intervals the Henon family possesses a homoclinic tangency. Therefore, one should expect long parameter intervals where the Henon family is not structurally stable. Strong numerical support for this observation is provided.
Border-Collision Bifurcation for Piecewise Smooth One-Dimensional Maps
We examine bifurcation phenomena for one-dimensional maps that are piecewise smooth and depend on a parameter m. In the simplest case, there is a point c at which the map has no derivative (it has two one-sided derivatives). The point c is the border of two intervals in which the map is smooth. As the parameter m is varied, a fixed point (or periodic point) E? may cross the point c, and we may assume that this crossing occurs at m = 0. The investigation of what bifurcations occurs at m = 0 reduces to a study of a map fm depending linearly on m and two other parameters a and b. A variety of bifurcations occur frequently in such situations. In particular, Em may cross the point c, and for m < 0 there can be a fixed point attractor, and for m 0 there may be a period-3 attractor or even a three-piece chaotic attractor which shrinks to E0 as m tends to 0. More generally, for every integer k = 2, bifurcations from a fixed point attractor to a period-k attractor, a 2k-piece chaotic attractor, a k-piece chaotic attractor, or a one-piece chaotic attractor can occur for piecewise smooth one-dimensional maps. These bifurcations are called border-collision bifurcations. For almost every point in the region of interest in the (a,b)-space, we state explicitly which border-collision bifurcation actually does occur. We believe this phenomenon will be seen in many applications.
Border-collision Bifurcations Including 'period two to period three' for Piecewise Smooth Systems
We examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter m. In the simplest case there is a surface G in phase space along which the map has no derivative (or has two one-sided derivatives). G is the border of two regions in which the map is smooth. As the parameter m is varied, a fixed point Em may collide with the border G, and we may assume that this collision occurs at m = 0. A variety of bifurcations occur frequently in such situations, but never or almost never occur in smooth systems. In particular Em? may cross the border and so will exist for m < 0 and for m 0 but it may be a saddle in one case, say m < 0, and it may be a repellor for m 0. For m < 0 there can be a stable period two orbit which shrinks to the point E0 as m tends to 0, and for m 0 there may be a stable period 3 orbit which similarly shrinks to E0 as m tends to 0. Hence one observes the following stable periodic orbits: a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbits. We also see analogously stable period 2 to stable period p orbit bifurcations, with p = 5,11,52, or period 2 to quasi-periodic or even to a chaotic attractor. We believe this phenomenon will be seen in many applications.
Chaotic Attractors on a 3-torus and Torus Break-up
Two coupled driven Van der Pol oscillators can have three-frequency quasiperiodic attractors, which lie on a 3-torus. The evidence presented in this paper indicates that the torus is destroyed when the stable and unstable manifolds of an unstable orbit become tangent. Furthermore, no chaotic orbits lying on a torus were observed, suggesting that, in most cases, at least in the case of this system, orbits do not become chaotic before their tori are destroyed. To expedite the calculations, a method was developed, which can be used to determine if an orbit is on a torus, without actually displaying that orbit. The method, also described in this paper, was designed specifically for our system. The basic idea, however, could be used for studying attractors of other systems. Very few modifications of the method, if any, would be necessary when studying systems with the number of degrees of freedom equal to that of our Van der Pol system.
Period halving for xn+1 = mF(xn) where F has negative Schwarzian derivativeWe present an example of a one-parameter family of maps F (x; m) = mF(x) where the map F if unimodal and has a negative schwarzian derivative. We will show for our example that (1) some regular period-halving bifurcations do occur and (2) the topological entropy can decrease as the parameter m is increased.
Why Period-Doubling Cascades Occur: Periodic Orbit Creation Followed by Stability Shedding
Period-doubling cascades of attractors are often observed in low-dimensional systems prior to the onset of chaotic behavior. We investigate conditions which guarantee that some kinds of cascades must exist.
How Often do Simple Dynamical Processes Have Infinitely Many Coexisting Sinks?
This work concerns the nature of chaotic dynamical processes. Sheldon Newhouse wrote on dynamical processes (depending on a parameter m) xn+1 = T(xn; m), where x is in the plane, such as might arise when studying Poincare return maps for autonomous differential equations in R3. He proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane, and that such parameter values m would be dense in some interval. The fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes. We prove, for an apparently typical situation, that the Newhouse construction yields only a set of parameter values m of measure zero.
Super Persistent Chaotic Transients
The unstable-unstable pair bifurcation is a bifurcation in which two unstable fixed points of periodic orbits of the same period coalesce and disappear as a system parameter is raised. For parameter values just above that at which unstable orbits are destroyed there can be chaotic transients. Then, as the bifurcation is approached from above, the average length of a chaotic transient diverges, and, below the bifurcation point, the chaotic transient may be regarded as having been converted into a chaotic attractor. It is argued that unstable-unstable pair bifurcations should be expected to occur commonly in dynamical systems. This bifurcation is an example of the crisis route to chaos. The most striking fact about unstable-unstable pair bifurcation crises is that long chaotic transients persist even for parameter values relatively far from the bifurcation point. These long-lived chaotic transients may prevent the time asymptotic state from being reach during experiments. An expression giving a lower bound for the average lifetime of a chaotic transient is derived and shown to agree well with numerical experiments. In particular, this bound on the average lifetime, T, satisfies T = k1 exp [k2 (a!a0)-1/ 2] for a near a0, where k1 and k2 are constants and a0 is the value of the parameter a at which the crisis occurs. Thus, as a approaches a0 from above, T increases more rapidly than any power of (a-a0)-1. Finally, we discuss the effect of adding bounded noise (small random perturbations) on these phenomena and argue that the chaotic transient should be lengthened by noise.
Period doubling cascades of attractors: A prerequisite for horseshoes
This paper shows that if a horseshoe is created in a natural manner as a parameter is varied, then the process of creation involves the appearance of attracting periodic orbits of all periods. Furthermore, each of these orbits will period double repeatedly, with those periods going to infinity.
Families of Periodic Orbits: Virtual Periods and Global Continuability
For a differential equation depending on a parameter, there have been numerous investigations of the continuation of periodic orbits as the parameter is varied. Mallet-Paret and Yorke investigated in generic situations how connected components of orbits must terminate. Here we extend the theory to the general case, dropping genericity assumptions.
Snakes: Oriented Families of Periodic Orbits, Their Sources, Sinks, and
Continuation
Poincare observed that for a differential equation dx/dt = f (x, a) depending on a parameter a, each periodic orbit generally lies in a connected family of orbits in (x,a)- space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: an orbit index J and a center index K defined at certain stationary points. We show that generically there are two types of Hopf bifurcation, those we call sources ( K = 1) and sinks ( K = -1). Generically if the set Q is bounded in (x, a) -space, and if there is an upper bound for periods of the orbits in Q, the Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter snake of orbits. A snake is a maximal path of orbits that contains no orbits whose orbit index is 0.
Crisis in Chaotic Scattering
We show that in a chaotic scattering system the stable and unstable foliations of isolated chaotic invariant sets can become heteroclinically tangent to each other at an uncountably infinite number of parameter values. The first tangency, whic h is a crisis in chaotic scattering, provides the link between the chaotic sets. A striking consequence is that the fractal dimension of the the set of singularities in the scattering function increases in the parameter range determined by the first and t he last tangencies. This leads to a proliferation of singularities in the scattering function and, consequently, to an enhancement of chaotic scattering.
Chaotic Scattering: An Introduction
In recent years chaotic behavior in scattering problems has been found to be important in a host of physical situations. Concurrently, a fundamental understanding of the dynamics in these physical situations has been developed, and such issues as symbolic dynamics, fractal dimensions, entropy, and bifurcations have been studied. The quantum manifestations of classical chaotic scattering is also an extremely active field, with new analytic techniques being developed and with experiments being c arried out. This issue of Chaos provides up-to-date survey of the range of work in this important field of study.
Quantum Manifestations of Chaotic Scattering
We examine chaotic scattering in the semiclassical regime for the two cases where the classical scattering is hyperbolic and nonhyperbolic. It is shown that in the nonhyperbolic case the energy dependent S-matrix autocorrelation functio n C() exhibits a cusp-shaped peak at
The Fractal Dimension in Nonhyperbolic Chaotic Scattering
In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic ( e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic base, although the singular set has zero total length, we present strong evid ence that its fractal dimension is 1.
Massive Bifurcation of Chaotic Scattering
In this paper we investigate a new type of bifurcation which occurs in the context of chaotic scattering. The phenomenology of this bifurcation is that the scattering is chaotic on both sides of the bifurcation, but, as the system parameter pa sses through the critical value, an infinite number of periodic orbits are destroyed and replaced by a new infinite class of periodic orbits. Hence the structure of the chaotic set is fundamentally altered by the bifurcation. The symbolic dynamics before and after the bifurcation, however, remains unchanged.
Transition to Chaotic Scattering
This paper addresses the question of how chaotic scattering arises and evolves as a system parameter is continuously varied starting from a value for which the scattering is regular (i.e., not chaotic). Our results show that the transition fro m regular to chaotic scattering can occur via a saddle-center bifurcation, with further qualitative changes in the chaotic set resulting from a sequence of homoclinic and heteroclinic intersections. We also show that a state of "fully developed" chaotic s cattering can be reached in our system through a process analogous to the formation of a Smale horseshoe. By fully developed chaotic scattering, we mean that the chaotic-invariant set is hyperbolic, and we find for our problem that all bounded orbi ts can be coded by a full shift on three symbols. Observable consequences related to qualitative changes in the chaotic set are also discussed.
Controlling Chaos
This lecture gives an intuitive understanding of chaos and its control through examples from different fields of research. The main goal of this presentation is to demystify these seemingly contradictory concepts: Chaos and Control.
Controlling Chaotic Fluctuations in Semiconductor Laser Arrays
A control scheme for eliminating the chaotic fluctuations observed in coupled arrays of semiconductor lasers driven high above threshold is introduced. Using the model equations, we show that the output field of the array can be stabilized to a steady in-phase state characterized by a narrow far-field optical beam. Only small local perturbations to the ambient drive current are involved in the control procedure. We carry out a linear stability analysis of the desired synchronized state and find that the number of active unstable modes that are controlled scales with the number of elements in the array. Numerical support for the effectiveness of our proposed control technique in both ring arrays and linear arrays is presented.
Preserving chaos: Control strategies to preserve complex dynamics with potential relevance to biological disorders
This paper considers the situation in which an originally chaotic orbit would, in the absence of intervention, become periodic as a result of slow system drift through a bifurcation. In the biological context, such a bifurcation is often desir able: there are many cases occurring in a wide variety of different situations, where loss of complexity and the emergence of periodicity are associated with pathology (such situations have been called "dynamical disease"). Motivated by this, we investiga te the possibility of using small control pertubations to preserve chaotic motion past the orbit where it would normally bifurcate to periodicity.
Controlling Chaos in a Temporally Irregular Environment
In this paper we examine the problem of controlling a chaotic system embedded in a time varying environment, where the environmental variation may be of relatively large amplitude, and may have a fairly irregular nature. Our results show that a previous method of controlling chaos, which selects and stabilizes unstable steady states or unstable periodic orbits, can be adapted to this time irregular situation, provided that one can make on-line, short-term predictions of the future evolution of the environment. We demonstrate this by using an example in which a ship is impacted by ocean waves on its side. The goal of control here is to prevent capsizing from taking place.
Synchronization of Spatiotemporal Chaotic Systems by Feedback Control
We demonstrate that two identical spatiotemporal chaotic systems can be synchronized by (1) linking one or a few of their dynamical variables, and (2) applying a small feedback control to one of the systems. Numerical examples using the diffus ively coupled logistic map lattice are given. The effect of noise and the limitation of the technique are discussed.
Converting Transient Chaos into Sustained Chaos by Feedback ControlA boundary crisis is a catastrophic event in which a chaotic attractor is suddenly destroyed, leaving a nonattracting chaotic saddle in its place in the phase space. Based on the controlling-chaos idea ["OGY"], we pre sent a method for stabilizing chaotic trajectories on the chaotic saddle by applying only small parameter perturbations. This strategy enables us to convert transient chaos into sustained chaos, thereby restoring attracting chaotic motion.
Using Small Perturbations to Control Chaos
The extreme sensitivity of chaotic systems to tiny perturbations (the butterfly effect) can be used both to stabilize regular dynamic behaviors and to direct chaotic trajectories rapidly to a desired state. Incorporating chaos deliberately into practical systems therefore offers the possibility of achieving greater flexibility in their performance.
Stabilizing Chaotic-Scattering Trajectories using Control
The method of stabilizing unstable periodic orbits in chaotic dynamical systems by Ott, Grebogi, and Yorke (OGY) is applied to control chaotic scattering in Hamiltonian systems. In particular, we consider the case of nonhyperbolic chaot ic scattering, where there exist Kolmogorov-Arnold-Moser (KAM) surfaces in the scattering region. It is found that for short unstable periodic orbits not close to the KAM surfaces, both the probability that a particle can be controlled and the average tim e to achieve control are determined by the initial exponential decay rate of particles in the hyperbolic component. For periodic orbits near the KAM surfaces, due to the stickiness effect of the KAM surfaces on particle trajectories, the average time to a chieve control can greatly exceed that determined by the hyperbolic component. The applicability of the OGY method to stabilize intermediate complexes of classical scattering systems is suggested.
Synchronization of Chaotic Trajectories using Control
We demonstrate that two identical chaotic systems can be made to synchronize by applying a small, judiciously chosen, temporal-parameter perturbations to one of them. This idea is illustrated with a numerical example. Other issues related to s ynchronization are also discussed.
High-Dimensional TargetingThis paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations. Such a proc edure is called "targeting". Previous work on targeting for chaotic attractors has been in the context of one- or two-dimensional maps. Here it is shown that targeting can also be done in higher dimensional cases. The method is demonstrated with a mechani cal system described by a four-dimensional mapping whose attractor has two positive Lyapunov exponents and a Lyapunov dimension of 2.8. The target is reached by making very small successive changes in a single control parameter. In one typical case, 35 it erates on average are required to reach a target region of diameter 10^(-4), as compared to roughly 10^(11) iterates without the use of the targeting procedure.
Controlling Hamiltonian Chaos
The method for stabilizing an unstable periodic orbit in chaotic dynamical systems originally formulated by Ott, Grebogi, and Yorke (OGY) is not directly applicable to chaotic Hamiltonian systems. The reason is that an unstable periodic orbit in such systems often exhibits complex-conjugate eigenvalues at one or more of its orbit points. In this paper we extend the OGY stabilization method to control Hamiltonian chaos by incorporating the notion of stable and unstable directions at each period ic point. We also present an algorithm to calculate the stable and unstable directions. Other issues specific to the control of Hamiltonian chaos are also discussed.
Using the Sensitive Dependence of Chaos to Direct Orbits to Targets in an Experimental Chaotic Systems
In this paper we present the first experimental verification that the sensitivity of a chaotic system to small perturbations (the "butterfly effect") can be used to rapidly direct orbits from an arbitrary initial state to an arbitrary accessib le desired state.
Using Chaos to Direct Orbits to Targets in Systems Describable by a One-Dimensional Map
The sensitivity of chaotic systems to small perturbations can be used to rapidly direct orbits to a desired state (the target). We formulate a particularly simple procedure for doing this for cases in which the system is describable by an approximately one-dimensional map, and demonstrate that the procedure is effective even in the presence of noise.
Controlling Chaos in High Dimensional Systems
Recently formulated techniques for controlling chaotic dynamics face a fundamental problem when the system is high dimensional, and this problem is present even when the chaotic attractor is low dimensional. Here we introduce a procedure for c ontrolling a chaotic time signal of an arbitrarily high dimensional system, without assuming any knowledge of the underlying dynamical equations. Specifically, we formulate a feedback control that requires modeling the local dynamics of only a single or a few of the possibly infinite number of phase-space variables.
Using Chaos to Target Stationary States of Flows
The sensitivity of chaotic systems to small perturbations is used to direct trajectories to a small neighborhood of stationary states of three-dimensional chaotic flows. For example, in one of the cases studied, a neighborhood which would typically take 1010 time units to reach without control can be reached using our technique in only about 10 of the same time units.
Controlling Chaotic Dynamical SystemsWe describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time perturbations of a control parameter. The time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor. The present paper extends that of "OGY", allowing for a more general choice of the feedback matrix and implementation to higher-dimensional sy stems. The method is illustrated by an application to the control of the periodically impulsively kicked dissipative dynamical system with two degrees of freedom resulting in a four-dimensional map (the "double rotor map"). A key issue addressed is that o f the dependence of the average time to achieve control on the size of the perturbations and on the choice of the control matrix.
Using Chaos to Direct Trajectories to TargetsA method is developed which uses the exponential sensitivity of a chaotic system to tiny perturbations to direct the system to a desired accessible state in a short time. This is done by applying a small, judiciously chosen, perturbation to an available system parameter. An expression for the time required to reach an accessible state by applying such a perturbation is derived and confirmed by numerical experiment. The method introduced is shown to be effective even in the presence of small-am plitude noise or small modeling errors.
Controlling ChaosIt is shown that one can convert a chaotic attractor to any one of a large number of possible attracting time-periodic motions by making only small time-dependent perturbations of an available system parameter. The method utilizes delay coordinate embedding, and so is applicable to experimental situations in which a priori analytical knowledge of the system dynamics is not available. Important issues include the length of a chaotic transient preceding the periodic motion, and th e effect of noise. These are illustrated with a numerical example.
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