Stability of equilibria of randomly perturbed maps

We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in ${\mathbb R}^d$. This condition can be used to stabilize weakly unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed linear and nonlinear maps in one- and two-dimensional spaces.

In this paper, we study the following difference equation in R d where q(x) = O(|x| 2 ) is a smooth function, A and B are deterministic and stochastic d × d matrices respectively. We assume that the spectral radius of A is slightly greater than 1, ρ(A) = 1 + , 0 < 1 and ask how to choose mean-zero matrix B = B( ) to stabilize the equilibrium at the origin. Our motivation for considering (3.1) is two-fold. On one hand, we want to understand how to tame weak instability in general d-dimensional maps by noise. Eventually, we want to apply these results to stabilize periodic orbits of randomly perturbed stochastic ordinary differential equations in R d+1 . In this case, (3.1) represents a Poincare map [20]. Stochastic stabilization of period orbits remains largely unexplored area of research with many promissing applications.
For scalar difference equations, stabilization was studied by Appleby, Mao, and Rodkina [6] and by Appleby, Berkolaiko, and Rodkina [4] (see also [2,11,3,9]). Certain higher-dimensional models similar to (3.1) were analyzed in the context of stability of finite-difference schemes (see [12] and references therein). In this paper, we show that one can achieve stability with high probability in a general d-dimensional model (3.1) under fairly general assumptions on B. The key requirement for stabilization is that matrix A −1 B must be diagonally dominant in the mean square sense.
The organization of this paper is as follows. In the next section, we prove a sufficient condition for stability (in probability) of an equilibrium in a d-dimensional map (cf. Theorem 2.4). To prove this theorem, we use the Strong Law of Large Numbers to show that the Lyapunov exponent of a typical trajectory is negative. The rest of the proof follows an argument developed for deterministic dynamical systems [23]. In §3 we apply Theorem 2.4 to the problem of stabilization. In §4, we illustrate our results with several numerical examples using one-and two-dimensional systems.

Stochastic stability
Consider an initial value problem for the following difference equation where (M n ) are independent copies of a d × d random matrix M ; q : for some C 1 , δ > 0. The initial condition x 0 is assumed to be deterministic.
Definition 2.1. [22] The equilibrium at the origin of (2.2) is said to be stable in probability if for any > 0 Then the equilibrium at the origin of (2.2) is stable in probability.
Remark 2.3. In (2.4), · is an arbitrary matrix norm. The same matrix norm is used throughout this section.
Condition (2.4) guarantees that the largest Lyapunov exponent of a generic trajectory is negative. This implies stability of x n ≡ 0 with high probability. Theorem 2.2 is a stochastic counterpart of the result of Koçak and Palmer for deterministic maps [23,Theorem 4]. It follows immediately from the proof of the following lemma, which also shows that the rate of convergence of (x n ) to the origin is exponential.
Thus, for every > 0 there exists n 0 such that In the remainder of the proof, we assume that 0 < < min{1, λ} is arbitrary but fixed. By (2.6), holds for all n > n 0 on the set of probability at least 1 − /2. From now on, we consider realizations (M k ) for which (2.7) holds.
Using (2.7), for any n > k ≥ n 0 , we have Further, for every 0 ≤ j < n 0 It then follows that on the set of probability at least 1 − , for all n ≥ k ≥ 1 we have Let 0 < η ≤ δ be arbitrary but fixed. Choose δ 1 > 0 such that Next, we prove that We prove (2.11) by induction. The statement in (2.11) clearly holds for i = 1 (cf. (2.2), (2.8), (2.9)). Assume that (2.11) holds for 0 ≤ i < p for p ≥ 1. We need to show that in this case (2.11) holds for i = p as well. To this effect, we note that as follows by iterating (2.2).
Using the triangle inequality, submultiplicativity of the operator norm, and (2.3), from (2.13) we obtain Here, we are also using the induction hypothesis, which implies that x j ∈ B δ , j = 0, 1, . . . , p − 1 so that (2.3) is applicable. Using (2.8) , we further derive (2.14) By applying the induction hypothesis to the quadratic term on the right-hand side of (2.14), for we have Using the discrete Gronwall's inequality (cf. Lemma 2.5), from (2.16) we have where we used (2.10) to derive the last inequality. Thus, |x p | ≤ ηµ p (cf. (2.15)).

Stabilization
Consider the following difference equation in R d : (B n ( )) are independent copies of a random matrix B( ) ∈ R d×d depending on a small parameter . We assume that the entries of B are mean zero (possibly dependent) random variables (RVs). Specifically, we let B = AG, where the entries (g ij ) of G are arbitrarily dependent, mean zero, non-degenerate RVs with finite third moments. In particular, we have where we have set σ 2 ij := var(g ij ), (i, j) ∈ [n] 2 . We also let σ := (σ 11 , σ 22 , . . . , σ nn ) and assume that lim →0 |σ( )| = 0.
We want to identify conditions on B that would stabilize the weakly unstable equilibrium in (3.1) with high probability. In the light of Theorem 2.4, stabilization will be achieved if Recall that it is sufficient to establish (3.4) in any matrix norm (see Remark 2.3). In the remainder of this section, we will use a matrix norm that satisfies where κ > 0 is arbitrary but fixed. Such norm always exists (cf. [21, Lemma 5.6.10]). We rewrite the second term on the right-hand side of (3.8) as follows where all off-diagonal terms of I + G are collected inG. By Gershgorin Theorem (cf. [21]), By the monotonicity of the logarithm, Taking expectations on both sides we get For each i By expanding the logarithm in the first term and using the fact that Eg ii = 0 we get Note that since log(1 + x) ≤ x, a bound on the big 'Oh' term will also give a bound on (3.9).
We estimate the terms above as follows Cauchy-Schwarz inequality and (3.7)) For m = 1, 2 and j = i, E|g i,j | m = o(σ 2 ij ) as verified above. Further, for 1 ≤ m ≤ 3 Hence, by (3.3) for all 1 ≤ i, j ≤ d, Plugging all of this into (3.8) and using ln(1 + + κ) ≤ + κ we obtain that This quantity can be made negative by choosing of κ = κ( ) sufficiently small and using (3.6)).

Examples
In this section, we illustrate our analysis of stabilization with several numerical examples.
as a sufficient condition for stabilization provided and σ are small enough. The results of numerical simulations of (4.1) with the linear map above with small positive initial condition are shown in Figure 1. Plot a shows that the trajectory of the random system with noise intensity subject to (4.2) after a brief explosion converges to the origin. The deterministic trajectory in b grows exponentially. Example 4.2. Next, we consider a nonlinear map f (x) = λx(1 − x). For λ = 1 + > 1, the logistic map f has two fixed points:x 1 = 0 andx 2 = (1 + ) −1 . For 0 < 1, the former is unstable, while the latter is stable. All trajectories of the deterministic map x → f (x) starting from x 0 ∈ (0, 1) converge tox 2 (see Fig. 2b). In the presence of noise, however, the iterations of (4.1) with high probability converge tox 1 , provided (4.2) holds and is small enough (see Fig. 2a).