The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and “cottage industries” have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution—a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (syste...

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Fundamentals of synchronization in chaotic systems, concepts, and applications.

Control of multistability

Part 1 Theory: probability elementary introduction to the theory of stable laws characteristic functions probability densities integral transformations special functions and equations multivariate stable laws simulations estimation Part 2 Applications: some probabilistic models correlated systems and fractals anomalous diffusion and chaos physics radiophysics astrophysics and cosmology stochastic algorithms financial applications miscellany Appendix

Chance and Stability: Stable Distributions and Their Applications

Second-order leader-following consensus of nonlinear multi-agent systems via pinning control

We consider three types of changes that attractors can undergo as a system parameter is varied. The first type leads to the sudden destruction of a chaotic attractor. The second type leads to the sudden widening of a chaotic attractor. In the third type of change, which applies for many systems with symmetries, two (or more) chaotic attractors merge to form a single chaotic attractor and the merged attractor can be larger in phase-space extent than the union of the attractors before the change. All three of these types of changes are termed crises and are accompanied by a characteristic temporal behavior of orbits after the crisis. For the case where the chaotic attractor is destroyed, this characteristic behavior is the existence of chaotic transients. For the case where the chaotic attractor suddenly widens, the characteristic behavior is an intermittent bursting out of the phase-space region within which the attractor was confined before the crisis. For the case where the attractors suddenly merge, the characteristic behavior is an intermittent switching between behaviors characteristic of the attractors before merging.In all cases a time scale \ensuremath{\tau} can be defined which quantifies the observed post-crisis behavior: for attractor destruction, \ensuremath{\tau} is the average chaotic transient lifetime; for intermittent bursting, it is the mean time between bursts; for intermittent switching, it is the mean time between switches. The purpose of this paper is to examine the dependence of \ensuremath{\tau} on a system parameter (call it p) as this parameter passes through its crisis value p=${p}_{c}$. Our main result is that for an important class of systems the dependence of \ensuremath{\tau} on p is \ensuremath{\tau}\ensuremath{\sim}\ensuremath{\Vert}p-${p}_{c}$${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ for p close to ${p}_{c}$, and we develop a quantitative theory for the determination of the critical exponent \ensuremath{\gamma}. Illustrative numerical examples are given. In addition, applications to experimental situations, as well as generalizations to higher-dimensional cases, are discussed. Since the case of attractor destruction followed by chaotic transients has previously been illustrated with examples [C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 57, 1284 (1986)], the numerical experiments reported in this paper will be for crisis-induced intermittency (i.e., intermittent bursting and switching).

http://yorke.umd.edu/Yorke_papers_most_cited_and_post2000/1987_14_Romeiras_GOY_PhysRevA_Crit_expon_crisis_intermit.pdf

Critical exponents for crisis-induced intermittency

An experimental study of the chaotic states and the routes to chaos in the driven pendulum as simulated by a phase-locked-loop electronic circuit is presented. For a particular value of the quality factor ($Q=4$), for which the chaotic behavior is found to be rich in structure, the state diagram (phase locked or unlocked) is established as a function of driving frequency and amplitude, and the nature of the chaos in these states is investigated and discussed in light of recent models of chaos in dynamical systems. The driven pendulum is found to exhibit symmetry breaking as a precursor to the period-doubling route for chaos. Although period doubling is found to be fairly common in the phase-locked states of the pendulum, it does not always manifest itself in complete bifurcation cascades. Intermittent behavior between two unstable phase-locked states is also commonly observed.

Chaotic states and routes to chaos in the forced pendulum

Comment on "Hopping Mechanism Generating 1f Noise in Nonlinear Systems"

On the observation of an uncompleted cascade in a Rayleigh-Bénard experiment

An experimental study of the flow of superfluid helium through a small hole (6-10 \ensuremath{\mu} diameter) between 0.3 and 1.7 \ifmmode^\circ\else\textdegree\fi{}K is reported. The hole is inside a small volume protected by a porous filter. Two distinct stable-flow regimes are observed: a fast velocity (up to 1000 cm/sec) which decreases as ${(\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}T})}^{\frac{1}{2}}$ between 0.5 and 1.7 \ifmmode^\circ\else\textdegree\fi{}K (in this case the dissipation mechanism is probaby the thermal excitation of vortex rings), and a slow temperature-independent velocity which increases as the square root of the chemical-potential difference, for which vortex emission from a source filament pinned in the flow is given as a model.

Critical velocities for superfluid-helium flow through a small hole

The focusing of acoustic waves is used to study nucleation phenomena in liquids. At large amplitude, non-linear effects are important so that the magnitude of pressure or density oscillations is difficult to predict. We present a calculation of these oscillations in a spherical geometry. 
We show that the main source of non-linearities is the shape of the equation of state of the liquid, enhanced by the spherical geometry. We also show that the formation of shocks cannot be ignored beyond a certain oscillation amplitude. The shock length is estimated by an analytic calculation based on the characteristics method. In our numerical simulations, we have treated the shocks with a WENO scheme. We obtain a very good agreement with experimental measurements which were recently performed in liquid helium. The comparison between numerical and experimental results allows in particular to calibrate the vibration of the ceramics used to produce the wave, as a function of the applied voltage.

D. D'Humieres

Papers

Chaotic states and routes to chaos in the forced pendulum

Comment on "Hopping Mechanism Generating 1f Noise in Nonlinear Systems"

On the observation of an uncompleted cascade in a Rayleigh-Bénard experiment

Critical velocities for superfluid-helium flow through a small hole

Non-linear effects and shock formation in the focusing of a spherical acoustic wave : Numerical simulations and experiments in liquid helium