Theory

Control & synchronisation of chaos

Two seemingly contradictory facts that the chaos community learned in 1990: (1) chaotic trajectories can be stabilised with vanishingly small perturbations; (2) two chaotic systems can be made to follow each other perfectly. These are the foundations of chaos engineering.

Why control is easy

A strange attractor is densely filled with unstable periodic orbits (UPOs). At every transit through the neighbourhood of a target UPO the local linearised dynamics has a one-dimensional unstable direction. A tiny parameter kick redirects the trajectory along the stable manifold, and the orbit settles onto the UPO.

OGY (Ott-Grebogi-Yorke, 1990)

Build the Poincaré section near the target UPO, identify the stable and unstable eigendirections, and at each piercing compute the parameter adjustment Δp that projects the trajectory onto the stable manifold. Implementations: Hénon, Lorenz, electronic circuits.

target: unstable fixed point  x* = 1 − 1/r = 0.7436

waiting phase: iterate freely until the trajectory enters
                the tolerance band |x − x*| < 0.050 around x*.

control kick:  pick δr such that f(x, r + δr) = x*, i.e.
                  δr  =  x* / (x (1 − x))  −  r

result: orbit is pinned to the previously-unstable
fixed point with arbitrarily small parameter perturbations.

Pyragas time-delay feedback (1992)

u(t) = K · [x(t) − x(t − T)]

Feedback proportional to the difference between the current state and the same state one period T ago. If x(t) is on a period-T orbit, the feedback vanishes automatically; if it drifts, the feedback pushes it back. Robust and model-free; subject to the odd-number limitation (cannot stabilise orbits with an odd number of real Floquet multipliers above 1) without extensions.

Pecora-Carroll synchronisation (1990)

Split a chaotic system into a stable and an unstable subsystem, send the unstable subsystem’s output as a drive signal, let the receiver evolve its own copy of the stable subsystem driven by the signal. Pecora and Carroll proved that under mild conditions (conditional Lyapunov exponents of the receiver subsystem negative), the receiver state converges to the master state.

master:  dx/dt = σ(y − x),  dy/dt = x(ρ − z) − y,  dz/dt = x y − β z
slave:                       dy_s/dt = x · (ρ − z_s) − y_s,
                              dz_s/dt = x · y_s − β z_s

The slave uses the master's x as the drive signal. Conditional Lyapunov
exponents of the (y_s, z_s) sub-system are negative, so |y_master − y_slave|
decays exponentially to zero. The receiver "locks on" to the chaotic master
state. Decoders for chaos-based communication exploit this.

Generalised, phase, lag synchronisation

• Generalised sync (Rulkov et al., 1995): there exists a continuous map Φ such that y(t) → Φ(x(t)) for slave y driven by master x.
• Phase sync (Rosenblum et al., 1996): only the instantaneous phases align while amplitudes remain uncorrelated. Common for weakly coupled non-identical oscillators.
• Lag sync: y(t) ≈ x(t − τ) for a fixed τ > 0.
• Anticipated sync: y(t) ≈ x(t + τ); achievable in delay-coupled systems.

Master Stability Function

Pecora and Carroll (PRL 1998) showed that synchronisation stability on an arbitrary coupled network factors into (a) a single MSF computed once per dynamical system, and (b) the eigenvalues of the graph Laplacian. Lets you design synchronisable networks rationally.

Anti-control of chaos (chaotification)

Sometimes you want to add chaos to a system: feedback chaos into a servo loop to break symmetry, drive a non-chaotic plant chaotic for fast mixing in microfluidics, randomise communication waveforms.

Communication with chaos

Chaos shift keying: encode a bit as the choice of one of two chaotic systems. Chaos masking: add an information signal to a chaotic carrier, send the sum, recover at the receiver via Pecora-Carroll. Practical issues: noise tolerance, low data rates, sync failure.

See also: /applications/circuits, /applications/control.