Chaos in dynamical systems.

*(English)*Zbl 0792.58014
Cambridge: Cambridge University Press,. xii, 385 p. (1993).

This is a broad text on chaos written from a physicist’s point of view. No proofs are given. However definitions and considerations are mathematically rigorous. For a mathematician working in dynamical systems it is a useful survey over most fashionable chaos topics and physicist’s works on them. The book seems also very useful and it is prepared to be readible for beginners, but in such a case I would advice a simultaneous reading or listening to a basic mathematical course on dynamical systems.

The content of the book is as follows:

1. Introduction and overview. Hénon attractor, sensitive dependence on initial conditions, other basic examples and notions.

2. One dimensional maps. Tent and logistic maps, bifurcations of periodic orbits, Šarkovskii’s order, invariant measures: Perron- Frobenius operator, applications in higher dimensions: Lorenz attractor.

3. Strange attractors and fractal dimension. The box-counting dimension, generalized baker maps, in the case of positive measure fractals: uncertainty exponent (including Cantor sets in parameters), Hausdorff dimension, pointwise \(\lim\textstyle{\log \mu(B)\over \log \text{diam}(B)}\), spectrum of \(D_ q\) dimensions: \[ D_ q = {1 \over 1 - q} \lim_{\varepsilon \to 0} {\log I(q,\varepsilon)\over \log (1/\varepsilon)},\quad I(q,\varepsilon) = \sum_ i \mu(B_ i)^ q \] the sum over all cubes of a grid of unit size \(\varepsilon\) covering the attractor (including information dimension for \(q = 1\) and correlation dimension \(q = 2\)), example: Couette-Taylor experiment.

4. Dynamical properties of chaotic systems. The horseshoe map and symbolic dynamics, zeros of vector fields, stable, unstable spaces and manifolds, homoclinic intersections. Hyperbolic invariant sets and Kaplan-Yorke conjecture that Lyapunov dimension is equal to the information dimension. Controlling chaos: hitting by a chaotic trajectory a stable manifold.

5. Nonattracting chaotic sets. Fractal basin boundaries (union of stable manifolds of saddles, graphs of Weierstrass nowhere differentiable functions). Chaotic scattering (by high potential islands). Nonuniform hyperbolic, nonattracting sets (Hénon with escapes).

6. Quasiperiodicity. Frequency spectrum, Arnold tongues of “frequency lockings” for \(\theta \mapsto \theta + wk \sin \theta\) mappings of the circle.

7. Chaos in Hamiltonian systems. KAM-theory, strongly chaotic systems (without elliptic islands), billiards.

8. Chaotic transitions. Period doubling cascades, the intermittency transitions (cycle saddle-nodes: tunnel effect, homoclinic and heteroclinic tangency). Lorenz maps. Basin boundary metamorphoses for the Hénon example. Bifurcation to chaotic scattering when the energy of particles decreases.

9. Multifractals. The singularity spectrum \(f(\alpha)\) where \(\varepsilon^{-f(\alpha)} \approx \sharp \{\varepsilon\text{-cubes with }\mu \approx \varepsilon^ \alpha\}\) and a relation to the dimensions spectrum \(D_ q\) with the use of the Legendre transform. Special cases where \(\mu\) is related to Lyapunov exponents. Invariant measure as limit of measures on periodic orbits.

Quantum chaos.

The content of the book is as follows:

1. Introduction and overview. Hénon attractor, sensitive dependence on initial conditions, other basic examples and notions.

2. One dimensional maps. Tent and logistic maps, bifurcations of periodic orbits, Šarkovskii’s order, invariant measures: Perron- Frobenius operator, applications in higher dimensions: Lorenz attractor.

3. Strange attractors and fractal dimension. The box-counting dimension, generalized baker maps, in the case of positive measure fractals: uncertainty exponent (including Cantor sets in parameters), Hausdorff dimension, pointwise \(\lim\textstyle{\log \mu(B)\over \log \text{diam}(B)}\), spectrum of \(D_ q\) dimensions: \[ D_ q = {1 \over 1 - q} \lim_{\varepsilon \to 0} {\log I(q,\varepsilon)\over \log (1/\varepsilon)},\quad I(q,\varepsilon) = \sum_ i \mu(B_ i)^ q \] the sum over all cubes of a grid of unit size \(\varepsilon\) covering the attractor (including information dimension for \(q = 1\) and correlation dimension \(q = 2\)), example: Couette-Taylor experiment.

4. Dynamical properties of chaotic systems. The horseshoe map and symbolic dynamics, zeros of vector fields, stable, unstable spaces and manifolds, homoclinic intersections. Hyperbolic invariant sets and Kaplan-Yorke conjecture that Lyapunov dimension is equal to the information dimension. Controlling chaos: hitting by a chaotic trajectory a stable manifold.

5. Nonattracting chaotic sets. Fractal basin boundaries (union of stable manifolds of saddles, graphs of Weierstrass nowhere differentiable functions). Chaotic scattering (by high potential islands). Nonuniform hyperbolic, nonattracting sets (Hénon with escapes).

6. Quasiperiodicity. Frequency spectrum, Arnold tongues of “frequency lockings” for \(\theta \mapsto \theta + wk \sin \theta\) mappings of the circle.

7. Chaos in Hamiltonian systems. KAM-theory, strongly chaotic systems (without elliptic islands), billiards.

8. Chaotic transitions. Period doubling cascades, the intermittency transitions (cycle saddle-nodes: tunnel effect, homoclinic and heteroclinic tangency). Lorenz maps. Basin boundary metamorphoses for the Hénon example. Bifurcation to chaotic scattering when the energy of particles decreases.

9. Multifractals. The singularity spectrum \(f(\alpha)\) where \(\varepsilon^{-f(\alpha)} \approx \sharp \{\varepsilon\text{-cubes with }\mu \approx \varepsilon^ \alpha\}\) and a relation to the dimensions spectrum \(D_ q\) with the use of the Legendre transform. Special cases where \(\mu\) is related to Lyapunov exponents. Invariant measure as limit of measures on periodic orbits.

Quantum chaos.

Reviewer: J.H.Przytycki (Riverside)

##### MSC:

37Cxx | Smooth dynamical systems: general theory |

37B99 | Topological dynamics |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

37N99 | Applications of dynamical systems |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

81Q50 | Quantum chaos |