# Even More Abstracts

The papers are further subdivided into the following categories for your convenience, and if you wish, you can go back to the Papers Section.

## General Chaos

Dynamics of an H\'enon-Lozi Type Map
Submitted (2000)
M.A. Aziz-Alaoui, C. Robert, and C. Grebogi
We present and analyze a smooth version of the piecewise linear Lozi map. The principal motivation for this work is to develop a map, which is better amenable for an analytical treatment as compared to the H\'enon map and is one that still possesses the characteristics of a H\'enon-type dynamics.  This paper is a first step. It does the comparison of the Lozi map (which is a piecewise linear version of the H\'enon map) with the map  that we introduce. This comparison is done for fixed parameters and also through global bifurcation by changing a parameter. If $\varepsilon$ measures the degree of smoothness, we prove that, as $\varepsilon\goes 0$, the stability and the existence of the fixed points is the same for both maps. We also numerically compare the chaotic dynamics, both in the form of an attractor and of a chaotic saddle.

Rotation Intervals for Chaotic Sets
Proceedings of the Amer. math. Soc. 126, 12805(1998)
K. Alligood and J. A. Yorke
Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two-periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction.

Dynamical Properties of a Simple Mechnical System With a Large Number of Coexisting Periodic Attractors
Chaos, Solitons and Fractals 9, 171 (1998)
U. Feudel, C. Grebogi, L. Poon and J. A. Yorke
We study a simple mechanical system consisting of two rotors that possesses a large number (3000+) of coexisting periodic attractors. A complex fractal boundary separates these tiny islands of stability and their basins of attraction. Hence, the long term behavior is acutely sensitive to the initial conditions. This sensitivity combined with many periodic sinks give rise to a rich dynamical behavior when the systems is subjected to small amplitude noise. This dynamical behavior is of great utility, and this is demonstrated by using perturbations which are smaller than the noise level to gear and influence the dynamics toward a specific periodic behavior.

Pseudocircles, Diffeomorphisms, and Perturbable Dynamical Systems
Ergodic Theory and Dyn. Sys. 16, 1031 (1996)
J. Kennedy and J. A. Yorke
We construct an example of a C ? diffeomorphism on a 7-manifold which has an invariant set with an uncountable number of pseudocircle components. Furthermore, any diffeomorphism which is sufficiently close (in the C 1 metric) to the constructed map has a similar invariant set. We also discuss the topological nature of the invariant set.

Bubbling transition
Phys. Rev. E 54, 1346 (1996)
S.C. Venkataramani, B.R. Hunt, and E. Ott
Recently, physcically important examples of dynamical system that have a chaotic attractor embedded in an invariant submanifold have been pointed out, and the unusual dynamical consequences of this situation have been studied. As a parameter e psilon of the system is increased, a periodiic orbit embedded in the attractor on the invariant manifold can become unstable for perturbations transverse to the invariant manifold. This bifurcation is called the bubbling transition, and it can lead to the occurrence of a recently discovered, new kind of basin of attraction, called a riddled basin. In this paper we study the effects of noise asymmetry on the bubbling transition. We find that, in the presence of noise or asymmetry, the attract or is replaced either by a chaotic transient or an intermittently bursting time evolution, and we derive scaling relations, valid near the bubbling transition, for the characteristic time (i.e., the average chaotic transient lifetime or the average interb urst time interval) as a function of the strength of the asymmetry and the variance of the additive noise. We also present numerical evidence for the predicted scalings.

Map with more than 100 coexisting low-period periodic attractors
Phys. Rev. E 54, 71 (1996)
U. Feudel, C. Grebogi, B.R. Hunt, and J.A. Yorke
We study the qualitative behavior of a single mechanical rotor with a small amount of damping. This system may possess an arbitrarily large number of coexisting periodic attractors if the damping is small enough. The large number of stable orb its yields a complex structure of closely interwoven basins of attraction, whose boundaries fill almost the whole state space. Most of the attractors observed have low periods, because high period stable orbits generally have basins too small to be detect ed. We expect the complexity described here to be even more pronounced for higher-dimensional systems, like the double rotor, for which we find more than 1000 coexisting low-period periodic attractors.

Universal grazing bifurcations in impact oscillators
Phys. Rev. E 53, 134 (1996)
F. Casas, W. Chin, C. Grebogi, and E. Ott
We examine the bifurcations of a piecewise smooth map that captures the universal properties of impact oscillators near grazing. In particular, we study periodic orbits with one impact per period and the way they are involved in the grazing bi furcations. We also show some phenomena that these orbits exhibit at grazing for some families of parameter values.

Predictability in Time Series
Phys. Lett. A 209, 327 (1995)
L.W. Salvino, R. Cawley, C. Grebogi, and J.A. Yorke
We introduce a technique to characterize and measure predictability in time series. The technique allows one to formulate precisely a notion of the predictable component of given time series. We illustrate our method for both numerical and exp erimental time series data.

Characterization of on-off intermittent time series
Phys. Lett. A 207, 173 (1995)
S.C. Venkataramani, Jr., T.M. Antonsen, E. Ott, and J.C. Sommerer
We outline results from a model that incorporates the universal featureas of systems that display on-off intermittency, and present numerical experiments realizing these results. the particular results investigated are the scaling of the power spectral density and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold. We also report the effect of additive noise on these results.

Self-Organization and Chaos in a Fluidized Bed
Phys. Rev. Lett. 75, 2308 (1995)
C.S. Daw, C.E.A. Finney, M. Vasudevan, N.A. van Goor, K. Nguyen, D.D. Bruns, E.J. Kostelich, C. Grebogi, E. Ott, and J.A. Yorke
We present experimental evidence that a complex system of particles suspended by upward-moving gas can exhibit low-dimensional bulk behavior. Specifically, we describe large-scale collective particle motin referred to as slugging in an individual device known as a fluidized-bed. As gas flow increases from zero, the bulk motion evolves from a fixed point to periodic oscillations to oscillations intermittenly punctuated by "stutters," which become more frequent as the flow increase s further. At the highest flow tested, the behavior becomes extremely complex ("turbulent").

Bizarre Topology is Natural in Dynamical Systems
Bullettin of the Amer. Math. Soc. 32, 309 (1995)
J.A. Kennedy and J.A. Yorke
We describe an example of a C infinitely differentiable diffeomorphism on a 7-manifold which has a compact invariant set such that uncountably many of its connected components are pseudocircles. (Any 7-manifold will suffice.) Furthermore, any diffeomorphism which is sufficiently close (in the C 1 metric) to the constructed map has a similar invariant set, and the dynamics of the map on the invariant set are chaotic.

Grazing Bifurcations in Impact Oscillators
Phys. Rev. E 50, 4427 (1994)
W. Chin, E. Ott, H.E. Nusse, and C. Grebogi.
Impact oscillators demonstrate interesting dynamical features. In particular, new types of bifurcations take place as such systems evolve from a nonimpacting to an impacting state (or vice versa), as a system parameter varies smoothly. These b ifurcations are called grazing bifurcations. In this paper we analyze the different types of grazing bifurcations that can occur in a simple sinusoidally forced oscillator system in the presence of friction and a hard wall with which the impacts take plac e. The general picture we obtain exemplifies universal features that are predicted to occur in a wide variety of impact oscillator systems.

Intermingled basins and two-state on-off intermittency
Phys. Rev. E 52, 3313 (1995)
Y.C. Lai and C. Grebogi
We consider dynamical systems which possess two low-dimensional symmetric invariant subspaces. In each subspace, there is a chaotic attractor, and there are no other attractors in the phase space. As a parameter of the system changes, the larg est Lyapunov exponents transverse to the invariant subspaces can change from negative to positive: the former corresponds to the situation where the basins of attractors are intermingled, while the latter corresponds to the case where the system ex hibits a two-state on-off intermittency. The phenomenon is investigated using a physical example where particles move in a two-dimensional potential, subjected to friction and periodic forcing.

Pseudocircles in Dynamical Systems
Transactions of the Amer. Math. Soc. 343, 349 (1994)
J. Kennedy and J.A. Yorke
We construct and example of a C 4 map on a 3-manifold which has an invariant set with an uncountable number of components, each of which is a pseudocircle. Furthermore, any map which is sufficiently close (in the C 1-metric) to the constructed map has a similar set.

Experimental control of Chaos for Communication
Phys. Rev. Lett. 73, 1781 (1994)
S. Hayes, C. Grebogi, E. Ott and A. Mark.
The use of chaos to transmit information is demonstrated experimentally. The symbolic dynamics of a chaotic electrical oscillator is controlled to carry a prescribed message by use of extremely small perturbing current pulses.

Border-Collision Bifurcations: A Possible Explanation for Observed Bifurcation Phenomena
Phys. Rev. E 49, 1043 (1994)
H.E. Nusse, E. Ott, and J.A. Yorke
Recently physical and computer experiments involving systems describable by continuous maps that are nondifferentiable on some surface in phase space have revealed novel bifurcation phenomena. These phenomena are part of a rich new class of bi furcations which we call border-collision bifurcations. A general criterion for the occurrence of border-collision bifurcations is given. Illustrative numerical results, including transitions to chaotic attractors, are presented. These border-colli sion bifurcations are found in a variety of physical experiments.

Enhancing Synchronism of Chaotic Systems
Phys. Rev. E 49, 945 (1994)
M. Ding and E. Ott
Recent work has considered the situation where a state variable (or variables) of a chaotically evolving system is used as an input to a replica of part of the original system. It was found that the replica subsystem often synchronizes to the chaotic evolution of the original system, and it has been suggested that this phenomenon may be used for secure communications. In this paper we point out that exact synchronism may also occur for a large class of systems that are not replicas of part of the original system. This allows greater freedom in choosing synchronizer systems, and we discuss the possibility of using this freedom to choose synchronizer systems with improved performance. Two explicit examples illustrating this statement are given, one where the chaotic system consists of three autonomous differential equations, and the other where the chaotic system is a two-dimensional map.

How Often Are Chaotic Saddles Nonhyperbolic
Nonlinearity 6, 779 (1993)
Y-C. Lai, C. Grebogi, J.A. Yorke and I. Kan
In this paper, we numerically investigate the fraction of nonhyperbolic parameter values in chaotic dynamical systems. By a nonhyperbolic parameter value we mean a parameter value at which there are tangencies between some stable and unstable manifolds. The nonhyperbolic parameter values are important because the dynamics in such cases is especially pathological. For example, near each such parameter value, there is another parameter value at which there are infinitely many coexisting attractors. In particular, Newhouse and Robinson proved that the existence of one nonhyperbolic parameter value typically implies the existence of an interval (a Newhouse interval) of nonhyperbolic parameter values. We numerically compute the fraction of nonhyperbolic parameter values for the Henon map in the parameter range where there exist only chaotic saddles (i.e., nonattracting invariant chaotic sets). We discuss a theoretical model which predicts the fraction of nonhyperbolic parameter values for small Jacobians. Two-dimensional diffeomorphisms with similar chaotic saddles may arise in the study of Poincare return map for physical systems. Our results suggest that (1) nonhyperbolic chaotic saddles are common in chaotic dynamical systems; and (2) Newhouse intervals can be quite large in the parameter space.

A Geometric Mechanism for Antimonotonicity in Scalar Maps with Two Critical Points
Phys. Rev. E 48, 1676 (1993)
S.P. Dawson, C. Grebogi and H. Kocak
Concurrent creation and destruction of periodic orbits - antimonotonicity - for one-parameter scalar maps with at least two critical points are investigated. It is observed that if, for a parameter value, two critical points lie in an interval that is a chaotic attractor, then, generically, as the parameter is varied through any neighborhood of such a value, periodic orbits should be created and destroyed infinitely often. A general mechanism for this complicated dynamics for one-dimensional m ultimodal maps is proposed similar to the one of contact-making and contact-breaking homoclinic tangencies in two-dimensional dissipative maps. This subtle phenomenon is demonstrated in a detailed numerical study of a specific one-dimensional cubic map.

Chaos in a Double Pendulum
American J. Physics 60, 491 (1992)
T. Shinbrot, C. Grebogi, J. Wisdom and J. A. Yorke
A novel demonstration of chaos in the double pendulum is discussed. Experiments to evaluate the sensitive dependence on initial conditions of the motion of the double pendulum are described. For typical initial conditions, the proposed experiment exhibits a growth of uncertainties which is exponential with exponent L = 7.5 plus or minus 1.5 s-1. Numerical simulations performed on an idealized model give good agreement, with the value L = 7.9 plus or minus 0.4 s-1. The exponents are positive, as expected for a chaotic system.

Antimonotonicity: Inevitable Reversals of Period-doubling Cascades
Phys. Lett. A 162, 249 (1992)
S.P. Dawson, C. Grebogi, I. Kan, H. Kocak, and J.A. Yorke
In many common nonlinear dynamical systems depending on a parameter, it is shown that periodic orbit creating cascades must be accompanied by periodic orbit annihilating cascades as the parameter is varied. Moreover, reversals from a periodic orbit creating cascade to a periodic orbit annihilating one must occur infinitely often in the vicinity of certain common parameter values. It is also demonstrated that these inevitable reversals are indeed observable in specific chaotic systems.

Algebraic Decay and Phase-space Metamorphoses in Microwave Ionization of Hydrogen Rydberg Atoms
Phys. Rev. A 45, 8284 (1992)
Y.C. Lai, C. Grebogi, R. Blumel, and M. Ding
We study classically the microwave ionization of hydrogen atoms using the standard one-dimensional model. We find that the survival probability of an electron decays algebraically for long exposure times. Furthermore, as the microwave field strenght incre ases, we find that the asymptotic algebraic decay exponent can decrease due to phase-space metamorphoses in which new layers of Kolmogorov-Arnold-Moser (KAM) islands are exposed when KAM surfaces are destroyed. We also find that after such phase-space met amorphoses, the survival probability of an electron as a function of time can have a crossover region with different decay exponents. We argue that this phenonmenon is typical for open Hamiltonian systems that exhibit nonhyperbolic chaotic scattering.

Experimental Observation of a Strange Nonchaotic Attractor
Phys. Rev. Lett. 65, 533 (1990)
W.L. Ditto, M.L. Spano, H.T. Savage, S.N. Rauseo, J. Heagy, and E. Ott
Evidence is presented for the existence of a strange nonchaotic attractor in a two-frequency quasiperiodically driven, buckled, magnetoelastic ribbon experiment. Scaling behavior in the Fourier amplitude spectrum is observed in agreement with predicted scaling behavior for strange nonchaotic attractors. Dimension measurements also support the existence of a strange nonchaotic attractor.

Experimental Observation of a Strange Nonchaotic Attractor
Phys. Rev. Lett. 65, 533 (1990)
W.L. Ditto, M.L. Spano, H.T. Savage, S.N. Rauseo, J. Heagy, and E. Ott
Evidence is presented for the existence of a strange nonchaotic attractor in a two-frequency quasiperiodically driven, buckled, magnetoelastic ribbon experiment. Scaling behavior in the Fourier amplitude spectrum is observed in agreement with predicted scaling behavior for strange nonchaotic attractors. Dimension measurements also support the existence of a strange nonchaotic attractor.

Multifractal Properties of Snapshot Attractors of Random Maps
Phys. Rev. A 41, 784 (1990)
F.J. Romeiras, V. Grebogi, and E. Ott
We consider qualitative and quantitative properties of "snapshot attractors" of random maps. By a random map we mean that the parameters that occur in the map vary randomly from iteration to iteration according to some probability distribution . By a "snapshot attractor" we mean the measure resulting from many iterations of a cloud of initial conditions viewed at a single instant (i.e., iteration). In this paper we investigate the multifractal properties of these snapshot attracto rs. In particular, we use the Lyapunov number partition function method to calculate the spectra of generalized dimensions and of scaling indices for these attractors; special attention is devoted to the numerical implemenation of the method and the evalu ation of statistical errors due to the finite number of sample orbits. This work was motivated by problems in convection of particles by chaotic flud flows.

Noise Reduction: Finding the Simplest Dynamical System Consistent with the Data
Physica D 41, 183 (1990)
E. Kostelich and J.A. Yorke
A novel method is described for noise reduction in chaotic experimental data whose dynamics are low dimensional. In addition, we show how the approach allows experimentalists to use many of the same techniques that have been essential for the analysis of nonlinear systems of ordinary differential equations and difference equations.

Spatiotemporal Dynamics in a Dispersively Coupled Chain of Nonlinear Oscillators
Phys. Rev. A 39, 4835 (1989)
D.K. Umberger, C. Grebogi, E. Ott, and B. Afeyan
A one-dimensional chain of forced nonlinear oscillators is investigated. This model exhibits typical behavior in periodically forced, spatially extended, nonlinear systems. At low driving amplitudes characteristic domainlike structure appears accompanied by simple asymptoptic time dependence. Before reaching its final state, however, the chain behaves chaotically. The chaotic transients appear as intermittent bursts mainly concentrated at the domain walls. At higher driving, the chaotic transi ent becomes longer and longer until the time dependence apparently corresponds to sustained chaos with the chain state characterized by the absence of domainlike spatial structure.

Multiple Coexisting Attractors, Basin Boundaries, and Basic Sets
Physica D 32, 296 (1988)
P. Battelino, C. Grebogi, E. Ott, J.A. Yorke, and E.D. Yorke
Orbits initialized exactly on a basin boundary remain on that boundary and tend to a subset on the boundary. The largest ergodic such sets are called basic sets. In this paper we develop a numerical technique which restricts orbits to the boun dary. We call these numerically obtained orbits "straddle orbits". By following straddle orbits we can obtain all the basic sets on a basin boundary. Furthermore, we show that knowledge of the basic sets provides essential information on the structure of the boundaries. The straddle orbit method is illustrated by two systems as examples. The first system is a damped driven pendulum which has two basins of attraction separated by a fractal basin boundary. In this case the basic set is chaotic and appears to resemble the product of two Cantor sets. The second system is a high dimensional system (five phase space dimensions), namely, two coupled driven Van der Pol oscillators. Two parameter sets are examined for this system. In one these cases the basin bou ndaries are not fractal, but there are several attractors and the basins are tangled in a complicated way. In this case all the basic sets are found to be unstable periodic orbits. It is then shown that using the numerically obtained knowledge of the basi c sets, one can untangle the topology of the basin boundaries in the five-dimensional phase space. In the case of the other parameter set, we find that the basin boundary is fractal and contains at least two basic sets one of which is chaotic and the othe r quasiperiodic.

Lorenz Cross Sections of the Chaotic Attractor of the Double Rotor
Physica D 24, 263 (1987)
E. Kostelich and J.A. Yorke
A Lorenz cross section of an attractor with k>0 positive Lyapunov exponents arising from a map of n variables is the transverse intersection of the attractor with an (n-k)-dimensional plane. We describe a numerical proc edure to compute Lorenz cross sections of chaotic attractors with k>1 positive Lyapunov exponents and apply the technique to the attractor produced by the double rotor map, two of whose numerically computed Lyapunov exponents are positive and wh ose Lyapunov dimension is approximately 3.64. Error estimates indicate that the cross sections can be computed to high accuracy. The Lorenz cross sections suggest that the attractor for the double rotor map locally is not the cross product of two interval s and two Cantor sets. The numerically computed pointwise dimension of the Lorenz cross sections is approximately 1.64 and is independent of where the cross section plane intersects the attractor. This numerical evidence supports a conjecture that the poi ntwise and Lyapunov dimensions of typical attractors are equal.

Critical Exponent of Chaotic Transients in Nonlinear Dynamical Systems
Phys. Rev. Lett. 57, 1284 (1986)
C. Grebogi,E. Ott and J.A. Yorke
The average lifetime of a chaotic transient versus a system parameter is studied for the case wherein a chaotic attractor is converted into a chaotic transient upon collision with its basin boundary (a crisis). Typically the average lifetime T depends upon the system parameter p via T is proportional [p-p0]-g, where p0 denotes the value of p at the crisis and we call ? the critical exponent of the chaotic transient. A theory determining ? for two-dimensional maps is developed and compared with numerical experiments. The theory also applies to critical behavior at interior crises.

Scaling Behavior of Windows in Dissipative Dynamical Systems
Phys. Rev. Lett. 54, 1095 (1985)
J.A. Yorke, C. Grebogi, E. Ott, and L. Tedeschini-Lalli
Global scaling behavior for period-n windows of dynamical systems is demonstrated. This behavior should be discernible in experiments.

Attractors on an N-torus: Quasiperiodicity Versus Chaos
Physica D 15, 354 (1985)
C. Grebogi, E. Ott and J. A. Yorke
The occurrence of quasiperiodic motions in nonconservative dynamical systems is of great fundamental importance. However, current understanding concerning the question of how prevalent such motions should be is incomplete With this in mind, the types of attractors which can exist for flows on the N - torus are studied numerically for N = 3 and 4. Specifically, nonlinear perturbations are applied to maps representing N - frequency quasiperiodic attractors. These perturbations can cause the original N - frequency quasiperiodic attractors to bifurcate to other types of attractors. Our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows: N - frequency quasiperiodic attractors are the most common, followed by ( N - 1)- frequency quasiperiodic attractors,..., followed by period attractors. However, as the nonlinearity is further increased, N-frequency quasiperiodicity becomes less common, ceasing to occur when the map becomes noninvertible. Chaotic attractors are very rare for N = 3 for small to moderate nonlinearity, but are somewhat more common for N = 4. Examination of the types of chaotic attractors that occur for N = 3 reveals a rich variety of structure and dynamics. In particular, we see that there are chaotic attractors which apparently fill the entire N - torus (i.e., limit sets of orbits on these attractors are the entire torus); furthermore, these are the most common types of chaotic attractors at moderate nonlinearities.

Strange Attractors that are not Chaotic
Physica D 13, 261 (1984)
C. Grebogi, E. Ott, S. Pelikan and J.A. Yorke
It is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic. Here we use the work strange to refer to the geometry or shape of the attracting set, while the word chaotic refers to the dynamics of orbits on the attractor (in particular, the exponential divergence of nearby trajectories). We first give examples for which it can be demonstrated that there is a strange nonchaotic attractor. These examples apply to a class of maps which model nonlinear oscillators (continuous time) which are externally driven at two incommensurate frequencies. It is then shown that such attractors are persistent under perturbations which preserve the original system type (i.e., there are two incommensurate external driving frequencies). This suggests that, for systems of the type which we have considered, nonchaotic strange attractors may be expected to occur for a finite interval of parameter values. On the other hand, when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observe to be converted to a periodic or chaotic orbit. Thus we conjecture that, in general, continuous time systems which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space.

Odd Chaos
Physics Letter 87, 271 (1982)
T.Y. Li, M. Misiurewicz, G. Pianigiani and J.A. Yorke
The simplest chaotic dynamical processes arise in models that are maps of an interval into itself. Sometimes chaos can be inferred from a few successive data points without knowing the details of the map. Chaos implies knowledge of initial data is insufficient for accurate long term prediction.

No Division Implies Chaos
Transactions of the American Mathematical Society 273, 191 (1982)
T.Y. Li, M. Misiurewicz, G. Pianigiani and J.A. Yorke
Let I be a closed interval in R1 and let f be a continuous map on I. Let x0 in I and xi + 1 = f (xi) for i 0. We say there is no division for (x1, x2,....xn) if there is no a in I such that xj < a for all j even and xj < a for all j odd. The main result of this paper proves the simple statement: no division implies chaos. Also given here are some converse theorems, detailed estimates of the existing periods, and examples which show that, under our conditions, one cannot do any better.

Interval Maps, Factors of Maps, and Chaos
Tohoku Mathematical Journal 32, 177 (1980)
J. Auslander and J.A. Yorke
We investigate the dynamical properties of continuous maps of a compact metric space into itself. The notion of chaos is defined as the instability of all trajectories in a set together with the existence of a dense orbit. In particular we show that any map on an interval satisfying a generalized period three condition must have a nontrivial (uncountable) minimal set as well as large chaotic subsets. The nontrivial minimal sets are investigated by lifting to sequence spaces while the chaotic sets are investigated using factors, projections of larger spaces onto smaller ones