## Exploring Chaos: Theory And ExperimentThis book presents elements of the theory of chaos in dynamical systems in a framework of theoretical understanding coupled with numerical and graphical experimentation. The theory is developed using only elementary calculus and algebra, and includes dynamics of one-and two-dimensional maps, periodic orbits, stability and its quantification, chaotic behavior, and bifurcation theory of one-dimensional systems. There is an introduction to the theory of fractals, with an emphasis on the importance of scaling, and a concluding chapter on ordinary differential equations. The accompanying software, written in Java, is available online (see link below). The program enables students to carry out their own quantitative experiments on a variety of nonlinear systems, including the analysis of fixed points of compositions of maps, calculation of Fourier spectra and Lyapunov exponents, and box counting for two-dimensional maps. It also provides for visualizing orbits, final state and bifurcation diagrams, Fourier spectra and Lyapunov exponents, basins of attractions, three-dimensional orbits, Poincaré sections, and return maps. Please visit http://www.maths.anu.edu.au/~ |

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### Contents

XI | 34 |

XII | 37 |

XIII | 41 |

XIV | 44 |

XV | 51 |

XVI | 55 |

XVII | 60 |

XVIII | 65 |

XIX | 68 |

XX | 72 |

XXI | 78 |

XXII | 82 |

XXIII | 89 |

XXIV | 92 |

XXV | 96 |

XXVI | 100 |

XXVII | 103 |

XXVIII | 107 |

XXIX | 113 |

XXX | 117 |

XXXI | 120 |

XXXII | 125 |

XXXIII | 130 |

XXXIV | 134 |

XXXV | 138 |

XXXVI | 143 |

XXXVII | 147 |

XXXVIII | 149 |

XLVII | 174 |

XLVIII | 179 |

XLIX | 184 |

L | 188 |

LI | 193 |

LII | 199 |

LIII | 200 |

LIV | 206 |

LV | 208 |

LVI | 210 |

LVII | 211 |

LVIII | 213 |

LIX | 215 |

LX | 218 |

LXI | 222 |

LXII | 223 |

LXIII | 225 |

LXIV | 226 |

LXVI | 227 |

LXVII | 228 |

LXVIII | 229 |

LXIX | 230 |

LXX | 231 |

LXXI | 232 |

LXXIII | 233 |

LXXIV | 236 |

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### Common terms and phrases

amplitudes axes basin of attraction behaviour Bifurcation Diagrams Bifurcation Diagrams window calculation Cantor set capacity dimension Chaos for Java chaotic orbits complex composition map computation convergence critical value Cubic 3 map curves defined definition derivative differential equations displayed driven pendulum dynamical system eigenvalues example fact factor final state diagrams formula fractal frequency function gives graph Graphical Analysis window Henon map infinite number initial conditions initial value interval 0,1 investigate Iterate(2d logistic map Lorenz equations Lozi map Lyapunov dimension Lyapunov exponents maximum non-linear number of iterations observed orbital density pair parameter values period doubling period doubling cascade periodic orbits phase plane pixel Poincare section points discarded previous exercise result return map scaling scrollbar self-similarity sequence shown in figure simple smooth solution stable fixed point stable period strange attractor tangent bifurcation tent map tool two-dimensional map unimodal unstable orbits window of Chaos zero zoom

### Popular passages

Page 5 - Newton's second law of motion states that the rate of change of momentum of a body is proportional to the applied force and takes place in the direction of that force.

Page 6 - ... acting in nature, as well as the momentary positions of all things of which the universe consists, would be able to comprehend the motions of the largest bodies of the world and those of the smallest atoms in one single formula, provided it were sufficiently powerful to subject all data to analysis; to it, nothing would be uncertain, both future and past would be present before its eyes.

Page 5 - ... proportional to the product of their masses, inversely proportional to the square of the distance between them, and directed along the line joining them at any instant of time.

Page 6 - ... which from different points of view equally engage the attention of analysts, and the solution of which would be of the greatest interest for the progress of science, the commission- respectfully proposes to his Majesty to award the prize to the best memoir on one of the following subjects : — I.

Page 6 - ... the same formula the motions of the largest bodies in the universe, and those of the lightest atoms: nothing would be uncertain to it, and the future as well as the past would be present to its eyes.

Page 159 - N(C] /,0\ <f_=lim - *-<., (43) 0 e-0 ln(l/e) where N(e) is the minimum number of n-dimensional cubes of side e needed to cover the set. For standard Euclidean objects, such as a point, a curve, or an area, the capacity (or box-counting) dimension is integer-valued.

Page 10 - Lorenz equations are distilled, the focus of interest is on convective fluid motion driven by heating from below, such as might occur locally over warm terrain.

Page 27 - behavior that is deterministic, or is nearly so if it occurs in a tangible system that possesses a slight amount of randomness, but does not look deterministic.

Page 149 - Scientists know a fractal when they see one, but there is no universally accepted definition . . , The...

Page 5 - Peterson [30], p229. of all things of which the universe consists, and further that it is sufficiently powerful to perform a calculation based on these data. It would then include in the same formulation the motions of the largest bodies in the universe and those of the smallest atoms.