Specifically, the general form for a stochastically switching continuous-time dynamical system (flow) is:
dx
Figure 1.8 Illustration of the stochastic switching process. The time axis is divided into intervals of length τ. For each interval, sk(t) = 1 with probability pk and sk(t) = 0 with
probability 1−pk.
wherex(t)∈RN is the state of the system at timet,Fgives the rate at which the state vector
x(t)changes (dependent on the current state and the switching process), and s(t)∈ {0,1}M
is a constant binary vector sk= [sk
1, sk2, . . . , skM] for the time interval t∈[(k−1)τ, kτ). Put
simply, we discretize the time axis into intervals of length τ time units. For the interval [(k −1)τ, kτ) the i-th entry in the binary vector sk is 1 with probability pi, and 0 with
probability 1−pi. That is, after every τ time, we flip a weighted coin; if the coin is heads
(with probability p), si(t) = 1, otherwise si(t) = 0, and remains constant until the next
switching time. The switching vector sk can be thought of as a random sequence generated by the switching process, and the events in si are independent and identically distributed
(i.i.d). The switching process is illustrated in Fig. 1.8. This means that each successive event does not depend on preceeding events (a0 in thei-th entry has no effect on the value of the (i+ 1)-th entry). When time is instead iterated in discrete units, the general form of the switching system (map) is given by:
x(k+ 1) =F(x(k),s(k)), (1.7)
where x(k) ∈ RN gives the state of the system at iteration k, F updates the state vector
based on its value at the current iteration,x(k)and the switching process, ands(k)∈ {0,1}M
afterm iterations, such that s((m−1)k) = s((m−1)k+ 1) =· · ·=s(mk−2) =s(mk−1). Again, si(k) (that is, the i-th entry of vector s(k) at time step k) is 1 with probability pi,
and 0 with probability1−pi and the switching process is i.i.d.
The nature of the switching process, while analogous in both of our settings, is quite different from other implementations of stochasticity in dynamical systems in the literature. Our blinking systems switch between a finite set of deterministic regimes, and are effectively deterministic at each time step, but randomly switch to a different regime after some pre- determined length of time. This is markedly different from examples that have stochastic variables chosen from a distribution and continuous switching times. For example, the most common way to add randomness to a system is to introduce a stochastic variable ξ(t) which follows either a uniform distribution withξ ∈[0,1] or normal distribution with ξ∼N(0,1). Typically, this method of incorporating stochasticity is called “noise.” Driving noise and is capable of inducing various dynamical phenomena, including stochastic synchronization [2, 6, 5] and stochastic resonance [155, 8, 7], and is so common that there is an entire class of differential equations (Stochastic Differential Equations) named for themn. Another, differ- ent, type of switching process was detailed in [91] in which the switching is a Markov process with a finite number of states, and switching rate is not fixed. In the discrete-time setting, typically stochasticity is implemented by switching at each time-step between random ma- trices pulled from a finite (or infinite) set of state matrices. Traditionally, the literature has focused on linear maps, and the term “Linear Jump Systems” has been coined for this setting [41].
Systems (1.6) and (1.7) provide the general mathematical description for switching dynamical systems. Although dynamical network models are covered by these general equa- tions, it will improve readability to describe the general form for continuous- and discrete- time switching networks (as they appear throughout this dissertation). Mathematically, a
blinking network of ODEs is given by:
dx
dt =F(x(t),s(t)) +H(x, t) (1.8)
which is the general form (1.6) with an additional term, H(x, t) that is an arbitrary, po- tentially “blinking,” vector-valued function that represents the manner in which the network topology affects each node. Typically, this H(x, t) is simply diffusive coupling, and the equation for the dynamics reduce to:
dx
dt =F(x(t),s(t)) +L(t)⊗H(x), (1.9)
where ⊗ is Kronocker matrix multiplication and L(t) is the Laplacian matrix at time t. Unpacking this a little further, we assume that the network is composed of N nodes, and each node has n state equations. Vector x is organized x = [x1x2· · ·xN]T such that the
state variables for node i are stacked on the state variables for node i + 1. L(t) gives the time-varying Laplacian matrix, and H(x) projects the coupling onto the appropriate state equations. For example, to describe a network of x-coupled Lorenz oscillators with a switching topology the equations for the i-th node would be:
˙ xi =σ(yi −xi) + N P j=1 lij(t)(xj −xi), ˙ yi =xi(ρ−zi)−yi, ˙ zi =xiyi−βzi.
The general form for a discrete-time blinking network is:
xi(k+ 1) =F(xi(k),s(k)) + N
X
j=1
L(k)H(xj(k)), (1.10)
wherei= 1,2, . . . , N,Lis the Laplacian matrix for the network at time stepk, andHis an arbitrary vector-valued function that defines the coupling between nodes.
While it may seem unnecessary to introduce these two general systems separately, the treatment of the time variable in either case is crucial to the analysis of the respective system. Mathematical tools for flows and maps are not the same, and even when they overlap, the implementation is not identical. For this reason, they must be treated as separate, though not entirely independent, cases.
With this in mind, we now review existing methods in blinking continuous- and discrete- time dynamical systems.