...

R. M. May, Simple mathematical models with very complicated dynamics, Nature 261, 459-467 (1976).

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...NAME="76"> .{

For an outline of the mathematical theory of dynamical systems see D. V. Anosov, I. U. Bronshtein, S.Kh. Aranson, and V. Z. Grines, Smooth dynamical systems, in Encyclopaedia of Mathematical Sciences, Vol. 1, edited by D. V. Anosov and V. I. Arnold (Springer-Verlag: New York, 1988).

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...weather.{

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos.\ Sci. 20, 130-141 (1963).

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...solution.

The full definition of a linear system also requires that the sum of scalar products 32#32, where 33#33 and 34#34 are constants, is also a solution.

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...ergodic

V. I. Arnold and A. Avez, Ergodic problems of classical mechanics (W. A. Benjamin: New York, 1968).

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...equations.

V. I. Arnold, Ordinary differential equations (MIT Press: Cambridge, MA, 1973). Also see D. K. Arrowsmith and C. M. Place, Ordinary differential equations (Chapman and Hall: New York, 1982).

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...problem,''

S. Smale, Differential dynamical systems, Bull. Am. Math. Soc. 73, 747-817 (1967).

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...collisions.

The coefficient of restitution 8#8 is called the damping coefficient  in the Bouncing Ball program.

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...Let

For a discussion of the motion of a particle in a constant gravitational field see any introductory physics text such as R. Weidner and R. Sells, Elementary Physics, Vol. 1 (Allyn and Bacon: Boston, 1965), pp. 19-22.

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...NAME="1364"> .

A nice, brief introduction to the C programming language sufficient for most of the programs in this book is Chapter 1: A tutorial introduction, of B. W. Kernighan and D. M. Ritchie, The C Programming Language (Prentice-Hall: Englewood Cliffs, NJ, 1978), pp. 5-31.

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...solutions,

The subscript n to 298#298 is labeling two distinct periodic orbits. This is potentially confusing notation since we previously reserved this subscript to label different points in the same periodic orbit. In practice this notation will not be ambiguous since this label will be a binary index, the length of which determines the period of the orbit. Different cyclic permutations of this binary index will correspond to different points on the same orbit. A noncyclic permutation must then be a point on a distinct period n orbit. The rules for this binary labeling scheme are spelled out in section 2.12.

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...pick.

Some initial conditions do not converge to the attractor. For instance, any x belonging to an unstable periodic orbit will not converge to the attractor. Unstable orbits are, by definition, not attractors, so that almost any orbit near an unstable periodic orbit will diverge from it and head toward some attractor.

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...attractors.

Technically, the phase space of the quadratic map, 331#331, can be compactified thereby making the point at infinity a valid attractor.

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...340#340,

See reference [5] for more details.

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...physics.

Feigenbaum introduced the renormalization group approach of critical phenomena to the study of nonlinear dynamical systems. Additional early contributions to these ideas came from Cvitanovic, and also Collet, Coullet, Eckmann, Lanford, and Tresser. The geometric convergence of the quadratic map was noted as early as 1958 by Myrberg (see C. Mira, Chaotic dynamics (World Scientific: New Jersey, 1987)), and also by Grossmann and Thomae in 1977.

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...NAME="1776"> ,

For a review of geometric series see any introductory calculus text, such as C. Edwards and D. Penny, Calculus and analytic geometry (Prentice-Hall: Englewood Cliffs, NJ, 1982), p. 549.

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...orbits.

In section 4.6.2 we show there exists a close connection between the existence of an infinity of periodic orbits and the existence of a chaotic invariant set, not necessarily an attractor. The term ``chaos'' in nonlinear dynamics is due to Li and Yorke, although the current usage differs somewhat from their original definition (see T. Y. Li and J. A. Yorke, Period three implies chaos, Am. Math. Monthly 82, 985 (1975)).

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...motion.

For a well-illustrated exploration of the Lyapunov exponent in the quadratic map system see A. K. Dewdney, Leaping into Lyapunov space, Sci. Am. 265 (3), pp. 178-180 (1991).

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...unstable.

Technically, the system is ``structurally stable.'' See section 1.9 of Devaney's  book for more details. Hyperbolicity and structural stability usually go hand-in-hand.

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...interval.

A partition of phase space that generates a one-to-one correspondence between points in the limit set and points in the original phase space is known in ergodic theory as a generating partition. Physicists loosely call such a generating partition a ``good partition'' of phase space. 

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...Mathematica

Mathematica is a trademark of Wolfram Research Inc. For a brief introduction to Mathematica see S. Wolfram, Mathematica, a system for doing mathematics by computer (Addison-Wesley: New York, 1988), pp. 1-23.

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...midpoint.

From the Lorentz force law , a wire carrying a current I, in a magnetic field of strength B, is acted on by a magnetic force 582#582. If the current I in the wire varies sinusoidally, then so does the force on the wire. See D. J. Griffiths, An introduction to electrodynamics (Prentice-Hall: Englewood Cliffs, NJ, 1981), pp. 174-181.

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...string.

For a review of the linear theory for the vibrations of a stretched string see A. P. French, Vibrations and waves (W. W. Norton: New York, 1971), pp. 161-170.

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...oscillator.

The term 599#599 in equation (3.6) is a typical physicist's notation meaning the dot product of the vector, 600#600.

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...Write

If we write 687#687, and 688#688, then it is easy to show from Euler's identity that 689#689. Thus, the use of complex numbers is not essential in this calculation; it is merely a trick that simplifies some of the manipulations with the sinusoidal functions.

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...

The closure of a set X is the smallest closed set that is a superset of X.

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...HREF="node83.html#s381">3.8.1).

A proper cross section would be a manifold transverse to the flow in 741#741, i.e., a four manifold such as 742#742. The torus attractor arises from a Hopf bifurcation -a bifurcation from a fixed point to an invariant curve. In a cross section, the limit cycle is represented by a fixed point. At the transition to quasiperiodic motion, this fixed point loses stability and gives birth to an invariant circle, which is a cross section of the torus in the flow.

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...system.

The attracting torus for nonplanar string vibrations is actually a four torus, 747#747.

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...spectra;

All things are fair in love, war, and experimental physics. Methods to obtain a spectrum analyzer may include: (a) locating and ``nationalizing'' a spectrum analyzer from a nearby laboratory, or (b) locating and appropriating a spectrum analyzer on grounds of ``national security.'' Another option is to use an older model spectrum analyzer, such as a Tektronix 1L5 spectrum analyzer which plugs into an older model Tek scope.

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...matrix

The components of the map 882#882 should be written as 897#897, where the superscript indicates the dependent coordinate. It is a common convention, though, to suppress the 882#882 and mix the dependent variables with the coordinate functions so that 898#898.

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...map.

The suspension is defined globally, while the cross section for a Poincaré map is only defined locally. Thus, these two constructions are not completely complementary.

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...derivative

Some books call this the differential of f and   denote it by 967#967.

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...f:

See Marsden and Tromba [5] for the n-dimensional definition.

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...conservative.

Note that this definition of dissipative can include expansive systems. These will not arise in the physical examples considered in this book. See Problem 4.12.

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...

Informally, a map of a surface is orientation preserving  if the normal vector to the surface is not flipped under the map.

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...positive.

As previously mentioned, the crossing convention and this definition of the relative rotation rate are the opposite of those originally adopted by Solari and Gilmore [4].

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...

For our purposes, a lift is a suspension consisting of flow with a simple twist.

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...by

The braid linking matrix is equivalent to the orbit matrix and insertion array previously introduced by Mindlin et al. [1]. See Melvin and Tufillaro for a proof [17].

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...NAME="7024"> 

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