Singular fold with real noise

We study the effect of small real noise on the jump behavior near a singular fold point, which is an important step in understanding the burst-spike behavior in many biological models. We show by the theory of center manifolds and random invariant manifolds that if the order of the noise is high enough, trajectories essentially pass the fold point in the manner as though there is no noise.

In [5], we extended the classical geometric singular perturbation theory of Fenichel [7]: System (1.1) has a critical manifold of equilibria when = 0 given by f (x, y, 0) = 0. The system may also be written in 'slow' time τ = t: x =f (x, y, ) + F (θ τ ω, x, y, ), y =g(x, y, ) + G(θ τ ω, x, y, ), (1.2) giving, for = 0, y-dynamics on the critical manifold. We showed that the centerstable, center-unstable, and center manifolds of the critical manifold persist for small = 0, and that the dynamics thereon may be analyzed under the condition of normal hyperbolicity. In particular, the slow y equation is perturbed by small noise as a regular perturbation on the new random center manifold.
In this paper, we study one very common case called the fold point where normal hyperbolicity fails. At a fold point, one stable eigenvalue of the Jacobian ∂f ∂x crosses the imaginary axis transversally at the origin. The well-known phenomenon of relaxation oscillation is related to this, in which solutions move slowly towards a fold point, jump from the fold point to another stable branch of the slow manifold, follow the slow dynamics again until reaching another fold point, and jump again, etc., with possibly consequent periodic solutions. Besides the classical relaxation oscillation, the burst-spike behavior is also related to the jumping behavior near a fold point. The geometric theory near fold points for deterministic systems was established by Szmolyan, Krupa and Wechselberger (see [8,11]).
1.1. Generic fold. In a 2-dimensional slow-fast systeṁ x =f (x, y, ), y = g(x, y, ), (1.3) where x ∈ R, y ∈ R, and > 0 is very small, a point (x 0 , y 0 ) is called a generic fold if the following conditions hold: and Assume, without loss of generality, that |f (x, y, 0) = 0} be the critical manifold. The above assumptions imply that there is a neighborhood U of the origin such that (0, 0) is the only point in U ∩ S where ∂f ∂x vanishes. By the Implicit Function Theorem, S ∩ U can be represented as y = φ(x) and is approximately the parabola y = − fxx(0,0,0) fy(0,0,0) x 2 .
The assumption ∂ 2 f ∂x 2 (0, 0, 0) > 0 implies that the left branch which we denote by S a is attracting and the right branch which we denote by S r is repelling. The reduced flow on the critical manifold is determined by The assumption g(0, 0, 0) < 0 implies that restricted to a sufficiently small neighborhood U , the reduced flow on S a and S r is towards the fold point. Fenichel's theory implies that outside an arbitrarily small neighborhood V of (0, 0), for small > 0, the manifolds S a and S r perturb smoothly to locally invariant slow manifolds S a, and S r, , which under the current set-up are single solutions. For small ρ > 0 and an appropriate interval J ⊂ R, let be a section in U transverse to S a , and let ∆ out = {(ρ, y)|y ∈ R} be a section transverse to the fast fibers (See Figure 1). For the transition map π : ∆ in → ∆ out , the following result is proved in [8]: Proposition 1.1. There exists 0 > 0 such that the following assertions hold for ∈ (0, 0 ]: is a positive constant.
x Figure 1. Critical manifold , slow manifolds, and sections for the fold point y S a S r S a, S r, ∆ out ∆ in 1.2. Stochastic model. When noise is taken into account, there are at least two fairly different ways to proceed: Either with a system of stochastic differential equations where noise is related to Brownian motion and Ito integrals are involved, or with a system of random differential equations: dx dt =f (x, y, ) + σ 1 F (θ t ω, x, y, ), where θ t ω is a metric dynamical system over a probability space, modeling the evolution of the noise.
In this paper, we study the random system (1.8) with the method of random dynamical systems. The noise in our case is assumed to be uniformly C 1 small, allowing us to use the random invariant manifold and foliation theory developed in [3,4,5].
Let (Ω, F, P ) be a probability space and X be a Banach space. Let T = R or Z endowed with their Borel σ− algebra.
The concept of random dynamical system was formulated as a means to rigorously analyze systems modeling many phenomena that are subject to uncertainty or random influences in areas as widely ranging as physics, biology, climatology, economics, etc. Those influences may arise through stochastic forcing, uncertain parameters, random sources or inputs, and random boundary conditions. The canonical example is the solution operator for a random differential equation driven by real noise: [1]). Throughout this paper, we suppose these basic conditions hold whenever there is a random differential equation of the form (1.9). Here, (Ω, F, P) is the classical Wiener space, i.e., Ω = {ω : ω(·) ∈ C(R, R d ), ω(0) = 0} endowed with the open compact topology so that Ω is a Polish space, and P is the Wiener measure. The measurable dynamical system θ t on the probability space (Ω, F, P) is given by the Wiener shift (θ t ω)(·) = ω(t + ·) − ω(t), for t > 0. It is well-known that P is invariant and ergodic under θ t . This measurable dynamical system θ t is also called a metric dynamical system and models the noise in the system. Another important class of generators for random dynamical systems may be found in stochastic ordinary differential equations, for instance system (1.7).
In this paper, we study system (1.8), in which we take σ i as functions of : where α > 0 and β > 1. We make the following: Assumptions. f , g, F , and G are C r+1 in (x, y, ) for some r ≥ 2, and the C 1 norms (x, y, ) of F and G are uniformly bounded. Moreover, F , G are measurable in ω and C 0 in t for a.e. ω ∈ Ω. Furthermore, f and g satisfy (1.4), (1.5), and (1.6). Under the above conditions, the main theorem in [5] implies that outside an arbitrary small neighborhood V of (0, 0), the critical manifolds S a and S r perturb smoothly to locally invariant random slow manifolds S a, ,ω and S r, ,ω for sufficiently small = 0. Under the two-dimensional set-up, S a, ,θ t ω and S r, ,θ t ω are actually trajectories of (1.10). S a, ,ω and S r, ,ω are obtained as -sections of two-dimensional, locally invariant random manifolds M a,ω and M r,ω , respectively, of the extended systemẋ =f (x, y, ) + α F (θ t ω, x, y, ), (Though the -direction is nonhyperbolic, its triviality enables dealing with the above parts as hyperbolic.) By the main theorem of [5], there exist attracting and repelling center-like locally invariant random manifolds M a,ω and M r,ω , -sections of which give S a, ,ω and S r, ,ω .
Yet [5] does not tell anything about the behavior of M a,ω at the fold point because of the loss of normal hyperbolicity. This is the topic of the current paper. Of particular interests are to determine how the part S a, ,ω passes the fold point, and where are the nearby dynamics. We will show that for high enough order noise, the dynamics essentially persist as though there is no noise.

Main results.
Recall that U is a neighborhood of (0, 0), in which S ∩ U is approximately the parabola y = − fxx(0,0,0) fy(0,0,0) x 2 . Also view U as its natural extension to include the -direction in R 3 , thus being a small neighborhood of (0, 0, 0). We choose U sufficiently small so that g(x, y, ) < 0 for (x, y, ) ∈ U . By (1.4), (1.5) and (1.6), a rescaling of x, y, and t yields the following canonical form: with F, G and θ being modified in an obvious manner yet with the same properties as in Assumptions. Also modify the defining interval J in ∆ in so that it is a cross section. We study the new transition map π from ∆ in to ∆ out defined by the random flow of system (1.12). Our main result reads: Theorem 3. For α ≥ 1 and β ≥ 4 3 , there exists 0 > 0 such that the following assertions hold for ∈ (0, 0 ] and all ω: 1. The manifold (trajectory) S a, ,θ t ω passes through ∆ out at a point (ρ, h( , ω)), where h( , ω) = O( Proof of the main theorem. Consider the extended system: =0. (2.1) 2.1. Blow-up. In [8], the authors apply the method of blow-up borrowed from [6] to overcome the difficulty caused by nonhyperbolicity. The blow-up is a coordinate transformation in which the nonhyperbolic fold point (0, 0, 0) is blown-up to a twosphere, which can be analyzed by standard methods of dynamical systems. We follow the same blow-up transformation. Letting S 2 be the two-sphere, we define B = S 2 × [0, ρ], where the constant ρ > 0 will be determined later and related to 0 by 0 = ρ 3 . The blow-up transformation is a mapping Φ : 2) with (x,ȳ,¯ ) ∈ S 2 ,r ∈ [0, ρ]. We choose ρ so small that Φ(B) ⊂ U .
Denote by X the random vector field corresponding to system (2.1). Since X vanishes at the origin, and the blow-up transformation restricted to S 2 × (0, ρ] is a diffeomorphism, there exists a random vector fieldX on B such that DΦ(X) = X. We analyze the vector fieldX on B in three different charts: K 1 , K 2 and K 3 , which are obtained by settingȳ = 1,¯ = 1, andx = 1, respectively, in the blow-up transformation (2.2).
The blow-up transformation in charts K 1 , K 2 , and K 3 are given by 3) with coordinates (x k , r k , k ) ∈ R 3 for k = 1, 2, 3. We then have the following coordinate changes between these charts: Let κ ij be the change of coordinates from K i to K j . One has 1. κ 12 is given by 2. κ 21 is given by 3. κ 23 is given by 4. κ 32 is given by (2.9) We borrow notation from [8]:P denotes an object in the blow-up which corresponds to an object P in the original problem. IfP is described in one of the charts, then P i denotes the object in chart K i for i = 1, 2, 3.
LetX 0 =X| S 1 ×[0,ρ] . Figure 3 sketches the phase portrait ofX 0 , which is deterministic and will be verified later. On the invariant circle S 1 , there are four equilibria: p a , p r , q in , q out . These equilibria are hyperbolic for the flow on S 1 , p a and q out being attracting, and p r and q in being repelling. The points p a and p r are end points of the blown-up critical manifoldsS a andS r , which are lines of equilibria forX 0 . The existence of these four (and only these four) equilibria will be verified by analyzing in local charts. Inserting (2.4) into (2.1), we obtain in chart K 2 the following system: For α, β > 2 3 , a rescaled time t 2 = r 2 t yields: which can be viewed as a regular (random) perturbation of the following Riccati equation, for which we have dropped the subscript for readability: Note that, since we care only about the transition map for r > 0 ( > 0), it does not matter what time scale we use for the random flow. The following facts hold for system (2.11).
Proposition 2.1. ( From [10]) The Riccati equation (2.11) has the following properties (see Figure 4): 1. Every orbit has a horizontal asymptote y = y r , where y r depends on the orbit, such that y approaches y r from above as x → ∞. 2. There exists a unique orbit γ 2 with parametrization as (x, s(x)), x ∈ R, which is asymptotic to the left branch of the parabola x 2 − y = 0 as x → −∞. The orbit γ 2 has a horizontal asymptote y = −Ω 0 < 0 such that y approaches −Ω 0 from above as x → ∞. 3. The function s(x) has the asymptotic expansions 4. All orbits to the right of γ 2 are backward asymptotic to the right branch of the parabola x 2 − y = 0.

5.
Every orbit to the left of γ 2 has a horizontal asymptote y = y l > y r , where y l depends on the orbit, such that y approaches y l from below as x → −∞. -Ω 0 (Refer to the notation of P ,P , and P i .) It will turn out that the corresponding orbitγ to γ 2 is backward asymptotic to the equilibrium p a and forward asymptotic to the equilibrium p out on the equator of S 2 . Moreover, close toγ, the trajectory S a, ,θ t ω leaves a neighborhood of p a , passes near the upper half of the two-sphere and enters into a neighborhood of q out .
Let π 2 be the transition map from Σ in 2 to Σ out 2 determined by (2.10). Let q 0 be the intersection of γ 2 with Σ in 2 . We have Proposition 2.2. The transition map π 2 has the following properties: (ii) A neighborhood of q 0 is mapped diffeomorphically into a neighborhood of π 2 (q 0 ). Remark 2.1. In the second part in Proposition 2.2, we mean that for each fixed ω, a fixed neighborhood of q 0 is mapped diffeomorphically onto a neighborhood of π 2 (q 0 ) and all of these images, created by varying ω, are contained in a fixed small neighborhood of π 2 (q 0 ).

2.3.
Dynamics in chart K 1 . Chart K 1 is used to analyze the dynamics in a neighborhood containing p a and p r . Inserting (2.3) into system (2.1) and rescaling with t 1 = r 1 t, we obtain the following system in chart K 1 : G 1 (θ t1 ω, x 1 , r 1 , 1 ). (2.12) This system has two (deterministic) invariant subsystems: on the invariant subspace 1 = 0, the system reads and on the invariant subspace r 1 = 0, the system reads (2.14) The intersection of these two subspaces is the x 1 -axis, restricted to which there are two hyperbolic equilibria: x 1 = ±1, corresponding to p r and p a , respectively, on the equator S 1 . We focus on the more interesting dynamics near p a (x 1 = −1). For the subsystem (2.13), the nonhyperbolic equilibrium (x 1 , r 1 ) = (−1, 0) has a one-dimensional attracting center manifold S a,1 , corresponding to the critical manifold S a of the original system.

2.
There exist stable random invariant foliations of the phase space with base M a,1,ω and one-dimensional fibers. For any c < 2, one can choose ρ, δ small enough such that the contraction along each fiber during a time interval [0, T ] is stronger than e −cT .
3. The special orbit γ 2 in chart K 2 is transformed by κ 21 to N a,1 .
Let R 1 be the rectangle in Σ in 1 defined by |1 + x 1 | ≤ β 1 , where β 1 is small. The constants ρ, δ and β 1 are chosen such that M a,1,ω ∩ Σ in 1 ⊂ R 1 for any ω. For 0 <˜ < δ, let I a (˜ ) be the line In order to study the transition map π 1 , we need first to estimate how long a trajectory stays in D 1 . From the third equation of system (2.12), we have This establishes: Lemma 2.2. The transition time T of a solution of system (2.12) from a point p = (x 1 , ρ, 1 ) ∈ Σ in 1 to the point π 1 (p) ∈ Σ out 1 satisfies T = 2 3 uniformly for any ω.