This book discusses some scaling properties and characterizes two-phase transitions for chaotic dynamics in nonlinear systems described by mappings. The chaotic dynamics is determined by the unpredictability of the time evolution of two very close initial conditions in the phase space. It yields in an exponential divergence from each other as time passes. The chaotic diffusion is investigated, leading to a scaling invariance, a characteristic of a continuous phase transition. Two different types of transitions are considered in the book. One of them considers a transition from integrability to non-integrability observed in a two-dimensional, nonlinear, and area-preserving mapping, hence a conservative dynamics, in the variables action and angle. The other transition considers too the dynamics given by the use of nonlinear mappings and describes a suppression of the unlimited chaotic diffusion for a dissipative standard mapping and an equivalent transition in the suppression of Fermi acceleration in time-dependent billiards.
This book allows the readers to understand some of the applicability of scaling theory to phase transitions and other critical dynamics commonly observed in nonlinear systems. That includes a transition from integrability to non-integrability and a transition from limited to unlimited diffusion, and that may also be applied to diffusion in energy, hence in Fermi acceleration. The latter is a hot topic investigated in billiard dynamics that led to many important publications in the last few years. It is a good reference book for senior- or graduate-level students or researchers in dynamical systems and control engineering, mathematics, physics, mechanical and electrical engineering.
Posing the problems.- A Hamiltonian and a mapping.- A phenomenological description for chaotic diffusion.- A semi phenomenological description for chaotic diffusion.- A solution for the diffusion equation.- Characterization of a continuous phase transition in an area preserving map.- Scaling invariance for chaotic diffusion in a dissipative standard mapping.- Characterization of a transition from limited to unlimited diffusion.- Billiards with moving boundary.- Suppression of Fermi acceleration in oval billiard.- Suppressing the unlimited energy gain: evidences of a phase transition.
Edson Denis Leonel is a Professor of Physics at São Paulo State University, Rio Claro, Brazil. He has been dealing with scaling investigation since his Ph.D. in 2003, where the first scaling investigation in the chaotic sea for the Fermi-Ulam model was studied. His research group developed different approaches and formalisms to investigate and characterize the several scaling properties in a diversity types of systems ranging from one-dimensional mappings, passing to ordinary differential equations, cellular automata, meme propagations, and also in the time-dependent billiards. There are different types of transition we considered and discussed in these scaling investigations: (i) transition from integrability to non-integrability; (ii) transition from limited to unlimited diffusion; and (iii) production and suppression of Fermi acceleration. The latter approach involves the analytical solution of the diffusion equation. His group and he published more than 160 scientific papers in respected international journals, including three papers in Physical Review Letters. He is the Author of Scaling Laws in Dynamical Systems (2021) by Springer and Higher Education Press and two Portuguese books, one dealing with statistical mechanics (2015) and the other one dealing with nonlinear dynamics (2019), both edited by Blucher.
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