{'fraction_non_alphanumeric': 0.048943918426802624, 'fraction_numerical': 0.012454479242534596, 'mean_word_length': 4.408034659314691, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Analysis of discrete and hybrid stochastic systems by nonlinear contraction theory

We investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) contracting stochastic system is upper-bounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete noisy interactions.


I. INTRODUCTION
Contraction theory is a set of relatively recent tools that provide a systematic approach to the stability analysis of a large class of nonlinear dynamical systems [1], [2], [3], [4]. A nonlinear nonautonomous systemẋ = f (x, t) is contracting if the symmetric part of the Jacobian matrix of f is uniformly negative definite in some metric. Using elementary fluid dynamics techniques, it can be shown that contracting systems are incrementally stable, that is, any two system trajectories exponentially converge to each other [1].
From a practical viewpoint, contraction theory has been successfully applied to a number of important problems, such as mechanical observers and controllers design [5], chemical processes control [6], synchronization analysis [2], [7] or biological systems modelling [8].
Recently, contraction analysis has been extended to the case of stochastic dynamical systems governed by Itô differential equations [4]. In parallel, hybrid versions of contraction theory have also been developped [3]. A hybrid system is characterized by a continuous evolution of the system's state, and intermittent discrete transitions. Such systems are pervasive in both artificial (e.g. analog physical processes controlled by digital devices) and natural (e.g. spiking neurons with subthreshold dynamics) environments. This paper benefits from these recent developments, and provides an exponential stability result for discrete and hybrid systems governed by stochastic difference and differential equations. More precisely, we prove in section II and III that the mean square distance between any two trajectories of a discrete (respectively hybrid resetting) stochastic contracting system is upper-bounded by a constant after exponential transients. This bound can be expressed as function of the noise intensities and the contraction rates of the noise-free systems. In section IV, we briefly discuss a number of theoretical issues regarding our analysis. In section V, we study, using the previously developped tools, the synchronization of noisy nonlinear oscillators that interact by discrete noisy couplings. Finally, some future directions of research are indicated in section VI.
Notations The symmetric part of a matrix A is defined as A s = 1 2 A + A T . For a symmetric matrix A, λ min (A) and λ max (A) denote respectively the smallest and the largest eigenvalue of A. A set of symmetric matrices (A i ) i∈I is uniformly positive definite if ∃α > 0, ∀i ∈ I, λ min (A i ) ≥ α. Finally, for a process x(t), we note E x (·) = E(·|x(0) = x).

II. DISCRETE SYSTEMS
We first prove a lemma that makes explicit the initial "discrete contraction" proof (see section 5 of [1]). Note that a similar proof for continuous systems can be found in [9].
Lemma 1 (and definition): Consider two metrics M i = Θ T i Θ i defined over R ni (i = 1, 2) and a smooth function f : R n1 → R n2 . The generalized Jacobian of f in the metrics Assume now that f is contracting in the metrics (M 1 , M 2 ) with rate β (0 < β < 1), i.e.
Then for all u, v ∈ R n , one has where d M denotes the distance associated with the metric M (the distance between two points is defined by the infimum of the lengths in the metric M of all continuously differentiable curves connecting these points).
Proof Consider a C 1 curve γ : [0, 1] → R n1 that connects u and v (i.e. γ(0) = u and γ(1) = v). The M 1 -length of such a curve is given by Since f is a smooth function, f (γ) is also a C 1 curve, with The chain rule next implies that which leads to Choose now a sequence of curves (γ n ) n∈N such that Finally, by letting n go to infinity in the last inequality, one obtains the desired result.
Theorem 1 (Discrete stochastic contraction): Consider the stochastic difference equation where f is a R n × N → R n function, σ is a R n × N → R nd matrix-valued function and {w k , k = 1, 2, . . .} is a sequence of independent d-dimensional Gaussian noise vectors, with w k ∼ N (0, Q k ). Assume that the system verifies the following two hypotheses (H1) the dynamics f (a, k) is contracting in the metrics (M k , M k+1 ), with contraction rate β (0 < β < 1), and the metrics (M k ) k∈N are uniformly positive definite. (H2) the impact of noise is uniformly upper-bounded by a constant Let a k and b k be two trajectories whose initial conditions are given by a probability distribution p(x 0 ) = p(a 0 , b 0 ). Then for all k ≥ 0 where [·] + = max(0, ·). This implies in particular that for all k ≥ 0 Proof Let x = (a, b) T ∈ R 2n . We have by the triangle inequality (to avoid long formulas, we drop the second argument of f and σ in the following calculations) Let us examine the conditional expectations of the three terms of the right hand side • From (H1) and lemma 1 one has Replacing v k by its expression in terms of u k then yields Next, integrating the last inequality with respect to x leads to (3). Finally, (4) follows from (3) by remarking that Remark In the particular context of state-independent metrics, hypothesis (H2) is equivalent to the following simpler condition which leads to the following stronger result instead of (4)

III. HYBRID SYSTEMS
We have derived above the discrete stochastic contraction theorem for time-and state-dependent metrics, contrary to the context of continuous systems, where the state-dependentmetrics version of the contraction theorem is still unproved [4].
We now address the case of hybrid systems, but due to the current limitations of continuous stochastic contraction, only state-independent metrics will be considered.
For clarity, we assume in this paper constant dwell-times, although more elaborate conditions regarding dwell-times can be adapted from [3].
Consider the hybrid resetting stochastic dynamical system All the contraction properties below will be stated with respect to a uniformly positive definite time-varying metric For all k, the discrete part is stochastically contracting at kτ with rate β < 1 and bound C d , i.e.
For all k, the continuous part is stochastically contracting in ]kτ, (k + 1)τ [ with rate λ > 0 and bound C c , i.e. ∀a ∈ R n , ∀t ∈]kτ, (k + 1)τ [, Let a(t) and b(t) be two trajectories whose initial conditions are given by a probability distribution p( and Proof For all t ≥ 0, let u(t) = E a(t) − b(t) 2 M(t) and let us study the evolution of u(t) between kτ + and (k + 1)τ + .
Condition (ii) and theorem 2 of [4] yield Next, condition (i) and theorem 1 above yield Substituting (9) into (10) leads to . Then, similarly to the proof of theorem 1, we have v k+1 ≤ r 1 v k , and then v k ≤ r k Now, for any t ≥ 0, choose k = ⌊t/τ ⌋. Then which leads to the desired result after some algebraic manipulations.

B. Only the discrete part is contracting
Let us examine now the more interesting case when the continuous part is not contracting, more precisely when λ ≤ 0 in (8). For this, we shall need to revisit the proof of theorem 2 in [4].
Theorem 3 (Case λ = 0): Assume all the hypotheses of theorem 2 except that λ = 0 in (8). Then for all t ≥ 0 Proof As in the proof of theorem 2 in [4], let Lemma 1 of [4] is unchanged, yielding (see [4] for more details) where A is the infinitesimal operator associated with the process x(t) (see section 2.1.2 of [4] or p. 15 of [10] for more details).
Remarks Theorems 3 and 4 show that it is possible to stabilize an unstable system by discrete resettings. If the continuous system is indifferent (λ = 0), then any sequence of uniformly contracting resettings is stabilizing. However, it should be noted that the asymptotic bound C 2 → ∞ when β → 1. In contrast, if the continuous system is strictly unstable (λ < 0), then specific contraction rates (depending on the dwell-time and the "expansion" rate of the continuous system) of the resettings are required. Finally, note that in both cases, the asymptotic bounds C 2 and C 3 are increasing functions of the dwell-time τ .

A. Modelling issue: distinct driving noise
In the same spirit as [4], and contrary to previous works on the stability of stochastic systems [12], the a and b systems considered in sections II and III are driven by distinct and independent noise processes. This approach enables us to study the stability of the system with respect to variations in initial conditions and to random perturbations: indeed, two trajectories of any real-life system are typically affected by distinct realizations of the noise. In addition, this approach leads very naturally to nice results regarding the comparison of noisy and noise-free trajectories (see section IV-B), which are particularly useful in applications (see e.g. section V).
However, because of the very fact that the two trajectories are driven by distinct noise processes, we cannot expect the influence of noise to vanish when the two trajectories get very close to each other. As a consequence, the asymptotic bounds 2C/(1 − β) (for discrete systems) and C 1 , C 2 , C 3 (for hybrid systems) are strictly positive. These bounds are nevertheless optimal, in the sense that they can be attained (adapt the Ornstein-Uhlenbeck example in section 2.3.1 of [4]).

B. Noisy and noise-free trajectories
Instead of considering two noisy trajectories a and b as in theorem 1, we assume now that a is noisy, while b is noisefree. More precisely, for all k ∈ N a k+1 = f (a k , k) + σ(a k , k)w k+1 To show the exponential convergence of a and b to each other, one can follow the same reasoning as in the proof of theorem 1, with C is replaced by C/2. This leads to the following result Corollary 1: Assume all the hypothesis of theorem 1 and consider a noise-free trajectory b k and a noisy trajectory a k whose initial conditions are given by a probability distribution p(a 0 ). Then, for all k ∈ N

Remarks
• The above derivation of corollary 1 is only permitted by our choice of considering distinct driving noise processes for systems a and b (see section IV-A). • Based on theorems 2, 3 and 4, similar corollaries can be obtained for hybrid systems. • These corollaries provide a robustness result for contracting discrete and hybrid systems, in the sense that any contracting system is automatically protected against noise, as quantified by (12). This robustness could be related to the exponential nature of contraction stability.

V. APPLICATION: OSCILLATOR SYNCHRONIZATION BY DISCRETE COUPLINGS
Using the above developped tools, we study in this section the synchronization of nonlinear oscillators in presence of random perturbations. The novelty here is that the interactions between the oscillators occur at discrete time instants, contrary to many previous works devoted to synchronization in the state-space 1 [14], [7].
Specifically, consider the Central Pattern Generator (CPG) delivering 2π/3-phase-locked signals of section 5.3 in [7]. This CPG consists of a network of three Andronov-Hopf oscillators x i = (x i , y i ) T , i = 1, 2, 3. We construct below a discretecouplings version of this CPG.
At instants t = kτ, k ∈ N, the three oscillators are coupled in the following way (assuming noisy measurements) Between two interaction instants, the oscillators follow the uncoupled, noisy, dynamics 1 Discrete couplings are more frequent in the literature devoted to phase oscillators synchronization, where phase reduction techniques are used [13]. However, contrary to our approach, these techniques are only applicable in the case of weak coupling strenghs and small noise intensities. where We apply now the projection technique developped in [7], [4]. We recommend the reader to refer to these papers for more details about the following calculations.
Consider first the (linear) subspace M of the global state space (the global state is defined by Let V and U be two orthonormal projections on M ⊥ and M respectively and consider Since the mapping is linear, using Itô differentiation rule yields the following dynamics for with Remark that g d (0) = 0 and g c (0) = 0 (the last equality holds because of the symmetry of f : ∀x, f (Rx) = R(f (x))). Thus, 0 is a particular solution to the noise-free version of the hybrid stochastic system (13,14).
Let us now examine the contraction properties of equations (13) and (14).
We have first Second, Since V is an orthonormal projection, one then has λ max ∂gc ∂ ⌢ y s ≤ 1.
To conclude, observe that Define the phase-locking quality δ by Rx i+1 − x i 2 then one finally obtains after exponential transients.
A numerical simulation is provided in Fig. 1.

VI. PERSPECTIVES
We are now focusing on the following directions of research: • proving the state-dependent-metrics version of the continuous and hybrid stochastic contraction theorems, • developping more elaborate conditions on dwell-times, and also hybrid switched versions of the theorems, • applying the synchronization-by-discrete-couplings analysis to other types of coupled dynamical systems, • studying the robustness of hybrid controllers and observers against random perturbations (for instance, the discrete observer for inertial navigation developped in [16]).