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Random switching between vector fields having a common zero. (English) Zbl 1437.60044
Authors’ abstract: Let $$E$$ be a finite set, $${\{F_i\}_{i\in E}}$$ a family of vector fields on $${\mathbb{R}^d}$$ leaving positively invariant a compact set $$M$$ and having a common zero $${p\in M}$$. We consider a piecewise deterministic Markov process $${(X_t,I_t)}$$ on $${M\times E}$$ defined by $${\dot{X}_t = F^{I_t}(X_t)}$$ where $$I$$ is a jump process controlled by $$X:$$
$${P(I_{t+s}=j | (X_u, I_u)_{u\leq t})=a_{ij}(X_t )s + o(s)}$$ for $${i\neq j}$$ on $${\{I_t=i\}}$$. We show that the behaviour of $${(X,I)}$$ is mainly determined by the behaviour of the linearized process $${(Y,J)}$$ where $${\dot{Y}_t=A^{J_t} Y_t},$$ $$A^i$$ is the Jacobian matrix of $${F_i}$$ at $$p$$ and $$J$$ is the jump process with rates $${(a_{ij} (p))}$$. We introduce two quantities $$\Lambda^{-}$$ and $$\Lambda^{+}$$, respectively, defined as the minimal (resp., maximal) growth rate of $${\|Y_t\|}$$, where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process $${(\Theta, J)}$$ with $${\Theta_t=Y_t/\|Y_t\|}.$$ It is shown that $$\Lambda^{+}$$ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of $${(Y, J)}$$ and that under general assumptions $${\Lambda^{-}=\Lambda^{+}}.$$ We then prove that, under certain irreducibility conditions, $${X_t\to p}$$ exponentially fast when $${\Lambda^{+}<0}$$ and $$(X,I)$$ converges in distribution at an exponential rate toward a (unique) invariant measure supported by $${(M\setminus\{p\})\times E}$$ when $${\Lambda^{-}>0}.$$
Some applications to certain epidemic models in a fluctuating environment (based on the differential equation due to A. Lajmanovich and J. A. Yorke [Math. Biosci. 28, 221–236 (1976; Zbl 0344.92016)]) are discussed and illustrate our results.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 34A37 Ordinary differential equations with impulses 37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents 37A50 Dynamical systems and their relations with probability theory and stochastic processes 92D30 Epidemiology
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