Asymptotic behaviour of blinking (stochastically switched) dynamical systems Vladimir Belykh Mathematics Department Volga State Academy Nizhny Novgorod Russia Blinking dynamical system and averaged system Finite time properties: 1 example Asymptotic properties: 1 example Igor Belykh Mathematics Department Georgia State University Atlanta GA USA Martin Hasler Laboratory of nonlinear systems School of Computer and Communication Sciences Swiss Federal Institute of Technology Lausanne (EPFL) Switzerland Torino, Nov.5, 2010

Blinking dynamical system Time axis: t t+Q  t+  t+  In each time interval one of the 2 M systems is chosen at random. Random variables S i k : value of i-th component of s during the time interval from (k-1)  to k . Hypothesis: all S i k independent, value 1 with probability p, 0 with prob. 1-p Terminology: s k : k = 1,2,…: switching sequence

Blinking dynamical system Torino, Nov.5,  t s 1 (t) Switching signals:

Torino, Nov.5, 2010 Averaged system For fixed x: Asymptotic time average = Ensemble average (law of large numbers)

Conjecture Torino, Nov.5, 2010 If switching is fast enough, the solution of the blinking system follows the solutions of the averaged system forever? for almost all switching sequences, or only with high probability? can we prove this?

Blinking system: stochastic averaging Torino, Nov.5, 2010, Theorem: Under some weak hypotheses, for any  > 0, T > 0, the probability that converges to 1 as   0 Clearly: as  increases,  decreases and/or T increases the probability decreases. But: Can we get an explicit lower bound the probability? or, equivalently, an upper bound for  for a given prob. A.V.Skorokhod, F.C.Hoppensteadt, H. Salehi, « Random Perturbation Methods », Springer Series in Applied Mathematical Sciences, vol. 150, 2002

Example 1: Lorenz System Torino, Nov.5, 2010  = 10, b = 8/3 Blinking: r is switched between r- = 28 and r+ = 33, probability p= 0.5  mean value 30.5

Example 1: Lorenz System Torino, Nov.5, 2010 x2 vs. x1 black: r = 30.5, blue: r = 28, red: r = 33

Example 1: Lorenz System Torino, Nov.5, 2010 x1 vs. time r = 30.5 r = 28 r = 33

Example 1: Lorenz System Torino, Nov.5, 2010 averaged system blinking system,  =10 -2 blinking - averaged

Example 1: Lorenz System Torino, Nov.5, 2010 blinking – averaged,  =10 -4 blinking – averaged,  =10 -2 Reducing  reduces much less the difference between blinking and averaged Reducing  practically does not extend the time interval

General theorem for finite time Torino, Nov.5, 2010 Theorem: Under some weak hypotheses, for each  > 0, T > 0 with probability converging to 1 as the switching time  goes to zero: where and c 1, c 2 are constants composed of bounds and Lipschitz constants of F and 

General theorem for finite time Torino, Nov.5, 2010 Case a: decrease of  is slow with  Case b:  must decrease rapidly as T increases is a soft limit for the validity of the averaging approximation

Can blinking follow averaged forever? Torino, Nov.5, 2010 In general: NO Exception: Solution of averaged system converges to an asymptotically stable equilibrium point x ∞ x ∞ is also an equilibrium point of the blinking system. If x ∞ is also asymptotically stable for the blinking system, corresponding solutions of the averaged and the blinking system stay always close together x∞x∞

Can blinking follow averaged forever? Torino, Nov.5, 2010 More generally: Solution of the averaged system converges towards an attractor (or attracting set) A A is an invariant set for the blinking system Possible that the solution of the blinking systems starting at the same initial state converges to A But the two solutions eventually separate A

Technical assumption: Averaged system has Lyapunov function Torino, Nov.5, 2010 Function with 1) 2) 3)

Technical assumption: Averaged system has Lyapunov function Torino, Nov.5, 2010 Lyapunov function  solutions converge to attractor (or attracting set) A A

Torino, Nov.5, 2010 Example 2: Synchronization of Lorenz systems 5 diffusively coupled Lorenz systems: Can we choose the diffusion constant d and the switching time  such that the dynamical systems synchronize completely and globally, i.e. All links are turned on and off independently with probability p = 0.5  most of the time the network is not connected!

Example 2: Synchronization of Lorenz systems Torino, Nov.5, 2010 For a large enough coupling constant d the diagonal linear subspace is an attracting set of the averaged system Then, as Lyapunov function can be taken with a suitable positive definite matrix M The attracting set of the averaged system is invariant for the blinking system

Torino, Nov.5, 2010 Example 2: 5 coupled Lorenz systems, d = 30, no switching (averaged system) x 2 – x 1 x 3 – x 1 x 4 – x 1 x 5 – x 1

Torino, Nov.5, 2010 Example 2: 5 coupled Lorenz systems, d = 60, p = 0.5,  = 0.1 x 2 – x 1 x 3 – x 1 x 4 – x 1 x 5 – x 1 Time axis: 0  

Torino, Nov.5, 2010 Example 2: 5 coupled Lorenz systems, d = 60, p = 0.5,  = 5 x 2 – x 1 x 3 – x 1 x 4 – x 1 x 5 – x 1

Example 2: Synchronization of Lorenz systems Torino, Nov.5, 2010, Almost all solutions of the blinking system synchronize if the switching time is smaller than some threshold and the coupling constant is large enough The solutions of the blinking and the averaged system do not stay close forever, but they both synchronize for large enough coupling constant and small enough switching time. Exponential speed of synchronization is similar for averaged and blinking model. Observations:

General theorem for infinite time when the attactor of the averaged system is invariant under the blinking system Torino, Nov.5, 2010 Theorem: Under some weak hypotheses and if there is a Lyapunov function W and an attractor (or attracting set) A of the averaged system such that A is invariant under the blinking system and there is a constant  >0 such that for all solutions  (t) of the averaged system Then, if  is small enough such that almost all solutions of the blinking system converge to A and there is a constant c 4 such that