## Chaos, Fractals, and Noise: Stochastic Aspects of DynamicsThe first edition of this book was originally published in 1985 under the ti tle "Probabilistic Properties of Deterministic Systems. " In the intervening years, interest in so-called "chaotic" systems has continued unabated but with a more thoughtful and sober eye toward applications, as befits a ma turing field. This interest in the serious usage of the concepts and techniques of nonlinear dynamics by applied scientists has probably been spurred more by the availability of inexpensive computers than by any other factor. Thus, computer experiments have been prominent, suggesting the wealth of phe nomena that may be resident in nonlinear systems. In particular, they allow one to observe the interdependence between the deterministic and probabilistic properties of these systems such as the existence of invariant measures and densities, statistical stability and periodicity, the influence of stochastic perturbations, the formation of attractors, and many others. The aim of the book, and especially of this second edition, is to present recent theoretical methods which allow one to study these effects. We have taken the opportunity in this second edition to not only correct the errors of the first edition, but also to add substantially new material in five sections and a new chapter. |

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### Contents

III | 1 |

IV | 5 |

V | 9 |

VI | 13 |

VII | 17 |

VIII | 19 |

IX | 31 |

X | 37 |

LV | 251 |

LVI | 252 |

LVII | 258 |

LVIII | 261 |

LIX | 264 |

LX | 268 |

LXI | 270 |

LXII | 273 |

XII | 41 |

XIII | 47 |

XIV | 51 |

XV | 58 |

XVI | 65 |

XVII | 71 |

XVIII | 79 |

XIX | 83 |

XX | 85 |

XXI | 86 |

XXII | 88 |

XXIII | 95 |

XXIV | 100 |

XXV | 102 |

XXVI | 105 |

XXVII | 112 |

XXVIII | 123 |

XXIX | 125 |

XXX | 129 |

XXXI | 136 |

XXXII | 139 |

XXXIV | 144 |

XXXV | 153 |

XXXVI | 156 |

XXXVII | 165 |

XXXVIII | 172 |

XXXIX | 175 |

XL | 183 |

XLI | 189 |

XLII | 190 |

XLIII | 191 |

XLIV | 195 |

XLV | 199 |

XLVI | 205 |

XLVII | 210 |

XLVIII | 215 |

XLIX | 226 |

L | 232 |

LI | 241 |

LII | 244 |

LIII | 246 |

LIV | 247 |

LXIII | 277 |

LXIV | 280 |

LXV | 283 |

LXVI | 289 |

LXVII | 292 |

LXVIII | 295 |

LXIX | 300 |

LXX | 303 |

LXXI | 304 |

LXXII | 306 |

LXXIII | 311 |

LXXIV | 315 |

LXXV | 320 |

LXXVI | 327 |

LXXVII | 330 |

LXXVIII | 333 |

LXXIX | 335 |

LXXX | 344 |

LXXXI | 346 |

LXXXII | 351 |

LXXXIII | 355 |

LXXXIV | 359 |

LXXXV | 364 |

LXXXVI | 368 |

LXXXVII | 371 |

LXXXVIII | 378 |

LXXXIX | 386 |

XC | 388 |

XCI | 391 |

XCII | 393 |

XCIII | 397 |

XCIV | 405 |

XCV | 411 |

XCVI | 417 |

XCVII | 420 |

XCVIII | 425 |

XCIX | 432 |

449 | |

CI | 457 |

461 | |

### Other editions - View all

Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics Andrzej Lasota,Michael C. Mackey Limited preview - 2013 |

### Common terms and phrases

absolutely continuous arbitrary assume assumption asymptotically periodic asymptotically stable baker transformation behavior Boltzmann equation Borel measure bounded compact support completes the proof consequence consider constant constrictive continuous function continuous time systems Corollary cr-algebra defined definition denote derivative differential equations dyadic transformation dynamical system entropy ergodic exact example exists Figure finite measure Fokker-Planck equation following theorem formula Frobenius-Perron operator corresponding given Hille-Yosida theorem implies inequality infinitesimal operator initial condition interval iterates J—oo Jx Jx Koopman operator Lebesgue integral limit lower-bound function mapping Markov operator measurable function measure preserving measure space mixing nonnegative nonsingular obtain perturbation Pf(x precompact probabilistic properties Proposition prove random variables random vector Remark result right-hand side satisfies semidynamical system semigroup semigroup Pt}t>o sequence solution stationary density stochastic kernel stochastic process stochastic semigroup strong convergence subset supp sweeping tion trajectory unique verify Wiener process

### Popular passages

Page 451 - Lasota A, and Mackey MC (1989) Stochastic perturbation of dynamical systems: the weak convergence of measures, J.