 ### Recent publications and presentations: Multistability of twisted states in non-locally coupled Kuramoto-type models

Taras Gyrnik, Martin Hasler, Yuri Maistrenko, Chaos 22, 013114 (2012).

A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior.

#### Dynamics of a stochastically blinking system. Part I: Finite time properties

Martin Hasler, Vladimir Belykh and Igor Belykh

SIAM Journal on Applied Dynamical Systems, 2013, Vol. 12, No. 2, pp. 1007-1030 We consider a time-varying dynamical system, called a blinking system, obtained by switching rapidlyand stochastically between a finite number of non-varying dynamical systems. One expects that the trajectories of the blinking system “wiggle” around the trajectories of the average of the non-varying systems. One expects also, that blinking and averaged system trajectories follow each other the closer, the faster the switching takes place.

These conjectures are more or less true, but only in finite time. In general, the trajectories of the blinking system eventually break away from the trajectories of the averaged system. We formulate the problem precisely and give rigorous explicit bounds for the distance between the blinking and the averaged trajectories. As the blinking system is stochastic, the bounds concern probabilities.

Example: The result does not distinguish between different types of trajectories, in particular not between stable and unstable trajectories. It is even applicable to chaotic dynamical systems where all trajectories are unstable. Let us consider the Lorenz system: For sigma = 10, b = 8/3 and r between 28 and 33, the system is known to have    chaotic behavior.

We consider the blinking system obtained by rapidly switching back and forth between Lorenz systems with r = 28 and r = 33. At densely,but regularly spaced times switching takes place or not with probability 1/2. The associated average system is again a Lorenz system, with r = 30.5. Trajectories of the blinking system and the averaged system that start at the same initial state follow each other closely before drifting apart, as is illustrated in the follwing picture. Top: x-xomponent of a trajectory of the averaged system

Middlex-xomponent of a sample trajectory of the blinking system starting from the same initial state.

Bottom:  Blinkingaveraged. The difference remains small initially, but then the two trajectories drift apart

#### Dynamics of a stochastically blinking system. Part II: Asymptotic properties

Martin Hasler, Vladimir Belykh and Igor Belykh

SIAM Journal on Applied Dynamical Systems, 2013, Vol. 12, No. 2, pp. 1031-1084

The results of part I do not distinguish between different types of trajectories of a stochastically blinking dynamical system or of the corresponding averaged system. It is therefore not surprising, that they do not give any insight into the asmptotic behavior, as time t –> infinity, of the trajectories.

In this part, we suppose that the averaged system has one or more attractors. All or part of its trajectories converge to this or these attractor(s) The question is then under what conditions and in which sense the trajectories of the blinking system converge also to this, or these, attractor(s).

A simple case can be readily examined. Remember that a blinking system is obtained by switching between different dynamical systems. Let us call them the constituent systems. If all solutions of all constituent systems converge to one and the same equilibrium point, then so do the trajectories of the blinking system, irrespective of the switching. sequence. This is not difficult to prove, one just needs some not very restrictive uniformity property for the convergence of the constituent systens.

In general, however, much less is required. It is not necessary to impose conditions on the attractors of the constituent dynamical systems. We just need that switching is fast with respect to the dynamics of the constituent systems and that the attractor(s) of the averaged system is (are) an invariant set(s) of the constituent systems.

Under these conditions the following hold.

• If the averaged system has a single global attractor A, then almost all trajectories of the blinking system converge towards A.
• If the averaged system is multistable (several attractors), and if a trajectory of the averaged system converges to the attractor A, then a trajectory of the blinking system starting at the same initial state converges with high probability also towards A.

Even if the attractor(s) of the averaged system is (are) not invariant sets of the constituent systems for the blinking system, but switching is still fast with respect to their dynamics, still general and meaningful results can be obtained:

• If the averaged system has a single global attractor A, then with high probability the trajectories of the blinking system get close to A and remain close to A for a long time (without converging to A)
• If the averaged system is multistable and if a trajectory of the averaged system converges to the attractor A, then with high probability the trajectories of the blinking system starting from the same initial condition get close to A and remain close to A for a long time.

In the following figure the 4 types of asymptotic behavior of the blinking system are schematically represented. A solid black oval represents an attractor of the averaged system, a pink oval enclosing it represents a neighborhood of the attractor and a dotted black line a boundary of the regions of attraction of two different attractors. A bold line represents a typical trajectory of the blinking system and a thin black line an untypical trajectory, occurring with small probability In the paper, 4 main theorems are precisely formulated and rigorously proved. They correspond to the 4 cases of asymptotic behavior of the blinking system sketched in the above figure. Explicit bounds for the following quantities are given. Except for case 1 they are probabilities of undesirable events.

• Case 1. Single attractor of the averaged system, invariant set of the blinking system: Minimum switching speed that guarantees almost sure convergence of the blinking trajectories to the attractor of the averaged system. Lower bound on the exponential speed of attraction of the blinking trajectory to the attractor.
• Case 2. Multiple attractors of the averaged system, invariant sets of the blinking system: Probability  that a trajectory of the averaged system and a trajectory of the blinking system starting at the same state do not converge to the same attractor. On the other hand, if they converge to the same attractor, lower bound on the exponential speed of convergence of the blinking trajetory to the attractor
• Case 3. Single attractor of the averaged system, not an invariant set of the blinking system: Probability that a trajectory of the blinking system takes at least a certain time to reach a certain neighborhood of the attractor. Probability that at a given time the trajectory of the blinking system lies outside of that neighborhood.
• Case 4. Multiple attractors of the averaged system, not invariant sets of the blinking system: Probability that a trajectory of the blinking system first reaches a neighborhood of a different attractor than the one the trajectory of the averaged system starting from the same initial state converges to. Probability that a trajectory of the blinking system takes more than a certain time to reach a neighborhood of the attractor to which the trajectory of the averaged system starting from the same initial state converges. Probability that a trajectory of the blinking system remains in the neighborhood of an attractor less than a certain time.