Chaos Lab

Applications · Engineering / Control theory

Control & synchronisation of chaos

Counter-intuitively, chaos is often easy to control: nearby unstable periodic orbits inside the attractor mean a tiny well-timed perturbation can stabilise the system on any of them. Pecora and Carroll's 1990 discovery that two chaotic systems can synchronise opened up chaos-based communication.

OGY chaos control (Ott-Grebogi-Yorke, 1990)

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.

Pecora-Carroll synchronisation (1990)

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.

OGY (Ott-Grebogi-Yorke)

1990 method: identify an unstable periodic orbit, linearise near it, deliver tiny parameter kicks each time the orbit drifts. Stabilises chaos with arbitrarily small control input.

Pyragas time-delay feedback

Add a control term proportional to x(t) − x(t − T) where T is the period of the target orbit. Robust, model-free, but limited (odd-number limitation).

Pecora-Carroll synchronisation

Drive a second copy of a chaotic system with a single signal from a master. Under mild stability conditions, the slave's state converges to the master's. The trick is choosing the 'drive' signal correctly.

Generalised, phase, lag synchronisation

Weaker forms of sync (Rulkov, Rosenblum, Pikovsky) hold when systems are non-identical or coupled weakly. Master Stability Function (Pecora-Carroll, 1998) tells you when sync is stable on networks.

See also

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Quick quiz

Test yourself on control

8 multiple-choice questions. Pick an answer for each, then submit to see explanations.

  1. Q1.OGY control of chaos works because:

  2. Q2.Pyragas time-delay feedback uses:

  3. Q3.Pecora-Carroll synchronisation requires:

  4. Q4.Generalised synchronisation means:

  5. Q5.The Master Stability Function (Pecora-Carroll 1998) relates synchronisation stability to:

  6. Q6.Anti-control of chaos refers to:

  7. Q7.Phase synchronisation (Rosenblum et al. 1996):

  8. Q8.OGY was published in:

0 of 8 answered