Control of Homoclinic Chaos by Weak Periodic Perturbations
This monograph presents a reasonably rigorous theory of a highly relevant chaos control method: suppression?enhancement of chaos by weak periodic excitations in low-dimensional, dissipative and non-autonomous systems. The theory provides analytical estimates of the ranges of parameters of the chaos-controlling excitation for suppression?enhancement of the initial chaos.The important applications of the theory presented in the book include: (1) control of chaotic escape from a potential well; (2) suppression of chaos in a driven Josephson junction; (3) control of chaotic solitons in Frenkel?Kontorova chains; (4) control of chaotic breather dynamics in perturbed sine-Gordon equations; (5) control of chaotic charged particles in electrostatic wave packets.
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A0max adiabatic adiabatic invariant aforementioned amplitude basin bifurcation biharmonic boundary function Chac6n chaos by weak chaos-controlling excitation chaos-inducing excitation chaos-suppressing excitation chaotic attractor chaotic escape Chap Cmax Cmin consider control of chaos corresponding damped differential equations dissipative dissipative systems driving dynamics effect exists Figure force frequency given by Eq Hamiltonian systems harmonic excitations homoclinic bifurcation homoclinic chaos homoclinic orbit homoclinic/heteroclinic initial conditions initial phase difference integer interval Jacobian elliptic function Josephson junction leading Lyapunov exponent Lemma Lett Lyapunov exponent main resonance Melnikov function Melnikov's method noise non-autonomous nonlinear normalized escape probability optimal parametric excitation particles phase space Phys Qopt range ratchet region resonance condition respectively satisfied separatrix shape parameter sine-Gordon equation soliton spatio-temporal straightforwardly obtains strange non-chaotic attractors subharmonic subharmonic resonances sufficient condition suitable initial phases suppression of chaos Suppression Theorems symmetry two-well Duffing oscillator unperturbed velocity wave packet waveform