Stochastic Bifurcation of Pathwise Random Almost Periodic and Almost Automorphic Solutions for Random Dynamical Systems

In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynamical systems. The existence and bifurcation of random periodic (random almost periodic, random almost automorphic) solutions have been established for a one-dimensional stochastic equation with multiplicative noise.


Introduction
This paper is concerned with almost periodic and almost automorphic dynamics of random dynamical systems associated with stochastic differential equations driven by time-dependent deterministic forcing. We will first define pathwise random almost periodic solutions and almost automorphic solutions for such systems, which are special cases of random complete solutions and random complete quasi-solutions. We then study existence, stochastic pitchfork and transcritical bifurcation of these types of solutions for one-dimensional non-autonomous stochastic equations.
Almost periodic and almost automorphic solutions of deterministic differential equations have been extensively studied by many experts, see, e.g., [17,22,23,25,27,28,29,30,31,35,36,38,39] and the references therein. In particular, the ω-limit sets of such solutions have been investigated in [23,27,28,29,30,31,35]. However, as far as the author is aware, it seems that there is no result available in the literature on existence and stability of pathwise random almost periodic or almost automorphic solutions for stochastic equations. The first goal of the present paper is to introduce these concepts for random dynamical systems generated by non-autonomous stochastic equations. Roughly speaking, a pathwise random almost periodic (almost automorphic) solution is a random complete quasi-solution which is pathwise almost periodic (almost automorphic) (see Definitions 2.1 and 2.2 below). It is worth mentioning that a pathwise random almost periodic (almost automorphic) solution is actually not a solution of the system for a fixed sample path, and it is just a complete quasi-solution in the sense of Definition 2.1. In this paper, in addition to existence of pathwise random periodic (almost periodic, almost automorphic) solutions, we will also study the stability and bifurcation of these solutions. More precisely, we will investigate stochastic pitchfork bifurcation of the one-dimensional non-autonomous equation and transcritical bifurcation of the equation where λ and δ are constants, β : R → R is positive, and γ : R × R → R satisfies some growth conditions. The stochastic equations (1.1) and (1.2) are understood in the sense of Stratonovich integration.
In the deterministic case (i.e., δ = 0), these equations are classical examples for demonstrating pitchfork and transcritical bifurcation of fixed points. In the stochastic case with constant β = 1 and γ = 0, the stochastic bifurcation of stationary solutions and invariant measures of (1.1)-(1.2) has been studied in [1,2]. In the real noise case, the same problem was discussed in [37].
When β = 1 and γ is time-independent, the bifurcation of stationary solutions of (1.1)-(1.2) was examined in [1,3]. For the bifurcation of stationary solutions of (1.1) with additive noise, we refer the reader to [14]. It seems that the bifurcation problem of (1.1) and (1.2) has not been studied in the literature when β and γ are time-dependent. The purpose of this paper is to investigate this problem and explore bifurcation of pathwise random complete solutions including random periodic (almost periodic, almost automorphic) solutions. Actually, for time-dependent β and γ satisfying certain conditions, we prove the pathwise complete quasi-solutions of (1.1) undergo a stochastic pitchfork bifurcation at λ = 0: for λ ≤ 0, the zero solution is the unique random complete quasisolution of (1.1) which is pullback asymptotically stable in R; for λ > 0, the zero solution is unstable and two more tempered random complete quasi-solutions x + λ > 0 and x − λ < 0 bifurcate from zero, i.e., lim λ→0 x ± λ (τ, ω) = 0, for all τ ∈ R and ω ∈ Ω.
The tempered random attractor A λ of (1.1) is trivial for λ ≤ 0, and is given by : τ ∈ R, ω ∈ Ω}. If, in addition, β and γ are both T -periodic in time for some T > 0, then x − λ and x + λ are also T -periodic. In this case, we obtain pitchfork bifurcation of pathwise random periodic solutions of (1.1). It seems that the bifurcation of almost periodic and almost automorphic solutions is much more involved. Nonetheless, for γ = 0, we will prove if β is almost periodic (almost automorphic), then so are x − λ and x + λ . As a consequence, we obtain stochastic pitchfork bifurcation of pathwise random almost periodic (almost automorphic) solutions of (1.1) in this case. By similar arguments, we will establish stochastic transcritical bifurcation of pathwise random complete quasi-solutions of (1.2) at λ = 0. If γ = 0 and β is periodic (almost periodic, almost automorphic), we further establish the transcritical bifurcation of random periodic (almost periodic, almost automorphic) solutions of (1.2) (see Corollary 4.2 and Theorem 4.3). This paper is organized as follows. In the next section, we introduce the concepts of pathwise random almost periodic (random almost automorphic) solutions for random dynamical systems (for pathwise random periodic solutions, the definition can be found in [16,40]). We will also review some results regarding pullback attractors. In the last two sections, we prove stochastic pitchfork bifurcation and transcritical bifurcation for equations (1.1) and (1.2), respectively.
Let (X, d) be a complete separable metric space and (Ω, F, P, {θ t } t∈R ) be a metric dynamical system as in [1] . Given a subset A of X, the neighborhood of A with radius r > 0 is denoted by N r (A). A mapping Φ: R + × R × Ω × X → X is called a continuous cocycle on X over R and (Ω, F, P, {θ t } t∈R ) if for all τ ∈ R, ω ∈ Ω and t, s ∈ R + , the following conditions (i)-(iv) are satisfied: Such Φ is called a continuous periodic cocycle with period T if Φ(t, τ + T, ω, ·) = Φ(t, τ, ω, ·) for every t ∈ R + , τ ∈ R and ω ∈ Ω. Let D be a collection of some families of nonempty subsets of X: (2.1) We now define D-complete solutions for Φ.
Definition 2.1. Let D be a collection of families of nonempty subsets of X given by (2.1).
(ii) ξ is called a random almost periodic function if for every ω ∈ Ω and ε > 0, there exists l = l(ω, ε) > 0 such that every interval of length l contains a number t 0 such that (iii) ξ is called a random almost automorphic function if for every ω ∈ Ω and every sequence If ξ is a complete quasi-solution of Φ and is also a random periodic (random almost periodic, random almost automorphic) function, then ξ is called a random periodic (random almost periodic, random almost automorphic) solution of Φ.
Notice that pathwise random periodic solution was introduced in [40]. We here further extend the concepts of deterministic almost periodic and almost automorphic solutions to the stochastic case. Definition 2.3. Let x 0 ∈ X and E be a subset of X. Then x 0 is called a fixed point of Φ if Φ(t, τ, ω, x 0 ) = x 0 for all t ∈ R + , τ ∈ R and ω ∈ Ω. A fixed point x 0 is said to be pullback Lyapunov stable in E if for every τ ∈ R, ω ∈ Ω and ε > 0, there exists δ = δ(τ, ω, ε) > 0 such that If x 0 is not pullback Lyapunov stable in E, then x 0 is said to be pullback Lyapunov unstable in E.
A fixed point x 0 is said to be pullback asymptotically stable in E if it is pullback Lyapunov stable in E and for all τ ∈ R, ω ∈ Ω and x ∈ E, Definition 2.4. Let D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} be a family of nonempty subsets of X. We say D is tempered in X with respect to (Ω, F, P, {θ t } t∈R ) if there exists x 0 ∈ X such that for every c > 0, τ ∈ R and ω ∈ Ω, lim t→−∞ e ct d(D(τ + t, θ t ω), x 0 ) = 0.
If, further, there exists T > 0 such that then A is called a periodic attractor with period T .

Pitchfork bifurcation of stochastic equations
In this section, we discuss pitchfork bifurcation of the following one-dimensional stochastic equation with deterministic non-autonomous forcing: where τ ∈ R, t > τ , x ∈ R, λ and δ are constants with δ > 0. The function β : R → R in (3.1) is smooth and positive. In addition, we assume there exist β 1 ≥ β 0 > 0 such that The function γ : R × R → R in (3.1) is smooth and there exist two nonnegative numbers c 1 and c 2 with c 1 ≤ c 2 < β 0 such that Note that condition (3.3) implies that γ(t, 0) = 0 for all t ∈ R. Therefore, x = 0 is a fixed point of equation ( There is a classical group {θ t } t∈R acting on (Ω, F, P ) which is given by , for all ω ∈ Ω and t ∈ R.
It follows from [1] that (Ω, F, P, {θ t } t∈R ) is a parametric dynamical system and there exists a θ t -invariant setΩ ⊆ Ω of full P measure such that for each ω ∈Ω, In the sequel, we only considerΩ rather than Ω, and hence we will writeΩ as Ω for convenience.
Since the solution of (3.1) is measurable in ω ∈ Ω and continuous in initial data, we find that Φ given by (3.6) is a continuous cocycle on R over (Ω, F, P, {θ t } t∈R ). We will study the dynamics of Φ in this section.
Given a bounded nonempty subset I of R, we write I = sup{|x| : x ∈ I}. Let D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} be a family of bounded nonempty subsets of R. Recall that D is tempered if for every c > 0, τ ∈ R and ω ∈ Ω, Denote by D the collection of all tempered families of bounded nonempty subsets of R, i.e., In the next subsection, we consider the bifurcation problem of (3.1) when γ is absent. In this case, the stochastic equation (3.1) is exactly solvable which makes it possible for one to completely determine its dynamics. We will show the random complete quasi-solutions of (3.1) undergo a pitchfork bifurcation when λ crosses zero from below. When β is periodic (almost periodic, almost automorphic), we show the random periodic (random almost periodic, random almost automorphic) solutions have similar bifurcation scenarios. We finally investigate pitchfork bifurcation of (3.1) with γ satisfying (3.3).

Pitchfork bifurcation of a typical non-autonomous stochastic equation
This subsection is devoted to pitchfork bifurcation of (3.1) without γ. In other words, we consider the following non-autonomous stochastic equation: As in the deterministic case, equation (3.9) is exactly solvable. To find a solution of (3.9), one may introduce a new variable y = x −2 for x = 0. Then y satisfies For every t, τ ∈ R with t ≥ τ , ω ∈ Ω and y τ ∈ R, by (3.10) we get Therefore, the solution x of (3.9) is given by, for every t, τ ∈ R with t ≥ τ , ω ∈ Ω and x τ ∈ R, It follows from (3.11) that, for each t ∈ R + , τ ∈ R, ω ∈ Ω and x 0 ∈ R, By (3.4) and (3.12) we get, for every λ > 0 and x 0 > 0, That is, for every λ > 0 and x 0 > 0, we have Note that the right-hand side of (3.13) is well defined in terms of (3.2) and (3.4). Similarly, by (3.12) we obtain, for every λ > 0 and x 0 < 0, (3.14) Given λ > 0, τ ∈ R and ω ∈ Ω, define It is evident that for each λ > 0 and τ ∈ R, both x + λ (τ, ·) and x − λ (τ, ·) are measurable. We next prove that x + λ and x − λ are random complete quasi-solutions of equation (3.1).
Proof. We first verify (3.33). By (3.2), (3.5) and Fatou's lemma we find that, for every τ ∈ R and which along with (3.15) yields (3.33). Note that the rest of this theorem is an immediate consequence of (3.12), Lemmas 3.1 and 3.2. The details are omitted here.
Next, we consider pitchfork bifurcation of random periodic, random almost periodic and random almost automorphic solutions of (3.9). Let β : R → R be a periodic function with period T > 0.
Then by (3.15) we see that for each λ > 0 and ω ∈ Ω, both x + λ (·, ω) and x − λ (·, ω) are T -periodic. In other words, x + λ and x − λ are random periodic solutions of (3.9) in this case. Applying Theorem 3.3, we immediately get pitchfork bifurcation of random periodic solutions for (3.9). In the almost periodic case, we need the following lemma. Proof. Given τ ∈ R and ω ∈ Ω, denote by We first show that g given by (3.34) is a random almost periodic function. Since β is almost periodic, given ε > 0, there exists l = l(ω, ε) > 0 such that every interval of length l contains a t 0 such that , for all t ∈ R.  36) which shows that g(·, ω) is almost periodic for every fixed ω ∈ Ω. By (3.2) we have, for every τ ∈ R and ω ∈ Ω, It follows from (3.36)-(3.37) that for each fixed ω ∈ Ω, g(·, ω) − 1 2 is almost periodic. Then the almost periodicity of x ± (·, ω) follows from (3.15) immediately, and this completes the proof.
Analogously, for the almost automorphic case, we have the following results. Note that the right-hand side of (3.40) is well defined due to (3.39). By (3.38), (3.40) and the Lebesgue dominated convergence theorem, we get, for every τ ∈ R and ω ∈ Ω, and By (3.41) and (3.42) we find that x + λ is a random complete quasi-solution of (3.9). By a similar argument, one can verify x − λ is also a random complete solution. This completes the proof.
As a consequence of Theorem 3.3, Lemmas 3.4 and 3.5, we get the following pitchfork bifurcation of random periodic (almost periodic, almost automorphic) solutions of (3.9).
Proof. Since β is periodic (almost periodic, almost automorphic), by Lemmas 3.4 and 3.5 we know that for every λ > 0, the random complete quasi-solutions x + λ and x − λ given by (3.15) are periodic (almost periodic, almost automorphic). Then, by Theorem 3.3 we conclude the proof.

Pitchfork bifurcation of a general non-autonomous stochastic equation
In this subsection, we discuss pitchfork bifurcation of the stochastic equation (3.1) with a nonlinearity γ satisfying (3.3). We first establish existence of D-pullback attractors for a generalized system and then construct random complete quasi-solutions. The comparison principle will play an important role in our arguments.
This implies the uniqueness of tempered complete quasi-solutions of (3.48), and thus completes the proof.
We now prove existence of D-pullback attractors for equation (3.44). Proof. Let t ∈ R + , τ ∈ R, ω ∈ Ω, D ∈ D and x τ −t ∈ D(τ − t, θ −t ω). By the comparison principle, we find the solution x of (3.44) satisfies where y is the solution of the linear equation (3.48). Then by (3.49) we have It follows from (3.51) and (3.59)-(3.60) that, for every τ ∈ R and ω ∈ Ω, where ξ is the complete quasi-solution of (3.48) given by (3.51). On the other hand, by (3.59) and (3.60), there exists T = T (τ, ω, D) > 0 such that for all t ≥ T ,  has two tempered complete quasi-solutions x * and x * such that Ω} is the unique D-pullback attractor of Φ.
As a consequence of Theorem 3.11, we have the following pitchfork bifurcation of random periodic solutions of (3.1).
Theorem 3.12. Let T be a positive number such that β(t + T ) = β(t) and γ(t + T, x) = γ(t, x) for all t ∈ R and x ∈ R. If (3.2) and (3.3) hold, then random periodic solutions of (3.1) undergo a stochastic pitchfork bifurcation at λ = 0. More precisely: (i) If λ ≤ 0, then x = 0 is the unique random periodic solution of (3.9) which is pullback asymptotically stable in R. In this case, the equation has a trivial D-pullback attractor.
(ii) If λ > 0, then the zero solution loses its stability and the equation has two more random x ± λ (τ, ω) = 0, for all τ ∈ R and ω ∈ Ω. (3.98) In this case, equation (3.9) has a D-pullback attractor Proof. By Theorem 3.11, we only need to show that for each λ > 0, the tempered complete quasisolutions x + λ and x − λ in (3.81) are T -periodic. Note that x + λ and x − λ are defined by (3.67) and (3.68) with x * and x * being replaced by x + λ and x − λ , respectively. In the present case, by Lemma 3.7 we find that ξ given by (3. 51) is T -periodic. Then, by (3.67) and the periodicity of β and γ, we get for all τ ∈ R and ω ∈ Ω, which shows that x + λ is T -periodic. Similarly, one can verify that x − λ is also T -periodic. The details are omitted.

Transcritical bifurcation of stochastic equations
In this section, we discuss transcritical bifurcation of the one-dimensional non-autonomous stochastic equation given by where λ, δ and β are the same as in (3.1); particularly, β satisfies (3.2). However, in the present case, we assume the smooth function γ satisfies the following condition: there exist two nonnegative numbers c 1 and c 2 with c 1 ≤ c 2 < β 0 such that By (4.2) we have γ(t, 0) = 0 for all t ∈ R, and hence x = 0 is a fixed point of (4.1). We will first discuss transcritical bifurcation of (4.1) when γ is zero and then consider the case when γ satisfies (4.2). We will also study transcritical bifurcation of random periodic (random almost periodic, random almost automorphic) solutions of (4.1).

(4.4)
It follows from (4.4) that if x 0 > 0, then the solution x(t, τ, ω, x 0 ) is defined for all t ≥ τ . Similarly, if x 0 < 0, then the solution x(t, τ, ω, x 0 ) is defined for all t ≤ τ . Based on this fact, we will be able to study the dynamics of (4.3) for positive initial data as t → ∞ as well as the dynamics for negative initial data as t → −∞. In the pullback sense, this allows us to explore the dynamics of solutions with positive initial data as τ → −∞ or with negative initial data as τ → ∞. By (4.4) we get that, for each t ∈ R + , τ ∈ R, ω ∈ Ω and x 0 ∈ R, (4.5) By (3.4) and (4.5) we obtain, for every λ > 0 and x 0 > 0, Analogously, by (4.5) we obtain, for every λ < 0 and x 0 < 0, Given λ ∈ R, τ ∈ R and ω ∈ Ω, we set if λ < 0. (4.8) By (4.8) we see that for every fixed τ ∈ R, x λ (τ, ·) is measurable. By an argument similar to Lemma 3.1 one can verify that x λ is a tempered complete quasi-solution of (4.3).
By (4.8) we see that if β is a periodic function with period T > 0, then so is x λ (·, ω) for all ω ∈ Ω. By an argument similar to Lemmas 3.4 and 3.5, one can prove x λ (·, ω) is almost periodic (almost automorphic) provided β is almost periodic (almost automorphic). Based on this fact, we have the following results from Theorem 4.1.
If, in addition, β is a periodic function, then so is x λ for λ = 0.