The definition of global random attractors for random dynamical system (RDS) established by Arnold [1] were proposed by Crauel and Flandoli [2], Schmalfuss [3]. Whereafter, Crauel et al [4] introduced the notion of stochastic dynamical system (SDS) and the global random attractors associated with SDS. Furthermore, the assertion that global random attractors are uniquely determined by attracting deterministic compact sets in phase space was attained by Crauel in [5]. Invoking these theory, the existence of global random attractors for SDS and RDS related to a plenty of mathematical physics problems have been studied by many researchers, for example, Ref [6, 7] and the references therein.
Once the infinite dynamical system has global attractors, the study on the structure of this invariant set is an interesting problem. Essentially, the Hausdorff dimension is one of the few pieces of information to describe the structure of global attractor in the aspect of analytical analysis. With respect to deterministic situation, the approaches to obtain the finite estimation for Hausdorff dimension of global attractors were due to Temam [8], Chepyzhov and Vishik [9], Caraballo et al [10]. As for the estimation for Hausdorff dimension of global random attractors, according to extend the methods in deterministic case, Crauel and Flandoli [11], Schmalfuss [12] provided the way to estimate the Hausdorff dimension of global random attractors, however, the conditions that assumed in that kind method are too conservative. The Assumptions in treatment given by Debussche in [13] can be satisfied by many RDS and SDS induced from mathematical physics problems, nevertheless, the results on estimation for the Hausdorff dimension obtained by this method do not have the relationship with global Lyapunov exponents clearly. Invoking the property of ergodicity, Debussche [14] established the powerful procedure to estimate the Hausdorff dimension of global random attractors. Furthermore, the estimation derived by the method possesses a strongly relationship with global Lyapunov exponents. By using this method, abundant literatures treated the estimation for Hausdorff dimension of global random attractors for a plenty of mathematical physics problems, for instance [15, 16, 17, 18, 19] and the references therein.
There exist two views in the study on behavior of dynamical system, the “static” standpoint and the “dynamical” standpoint, which are not distinguished in the deterministic case, however, the clear connection between the two views in the stochastic status do not exists, for more detail, one can refer to [1, 20]. There exist some results on “dynamical” global dynamics on stochastic dynamical system, for example, Crauel and Flandoli [20] asserted that additive noise destroys pitchfork bifurcation in one dimensional system. The fact that parametric noise (even a multiplicative white noise) destroys Hopf bifurcation was proved by Arnold et al [21]. Wang [22] focused on the bifurcation for stochastic parabolic equations. The investigation on stochastic bifurcation in Duffing system by the theory of random attractors was due to Schenk-Hoppé[23].
Heat conduction process [24], the purpose of which is to determine the distribution of temperature in solid or static fluid, is a very important problem in practical applications, such as civil engineering, high-speed aircraft. Actually, the dynamical govern equation for heat conduction is one kinds of nonlinear stochastic reaction-diffusion equations, the results on global attractor for the system in deterministic case, see Ref [8] and [9]. Debussche [14] obtained the estimation for Hausdorff dimension of global random attractors of reaction-diffusion equations driven by additive white noise. As for the case that the system under multiplicative noise loading with a constant coefficient, the Hausdorff dimension was estimated by Caraballo et al [16]. Afterwards, Caraballo et al [25] studied the stochastic pitchfork bifurcation of the system, the main approach in which is based on finding some invariant manifold to derive the lower bounds on the dimension of global random attractors. It is a very wonderful work in the investigation on global dynamics of stochastic system, however, it is hardly understood by the engineering. Thus, in light of the Hasudorff dimension obtained by Fan and Chen [18], the global dynamics of stochastic heat conduction with multiplicative white noise with a variable coefficient are studied numerically by employing the stochastic subdivision algorithm method proposed by Keller and Ochs [26] to achieve the global random attractors. To our best knowledge, there hardly exist such investigation on this problem.
The rest content of this paper is organized as follows. In Section 2, the main preliminaries related to existence and Hausdorff dimension of the stochastic heat conductio are given. Section 3 is intended to provided the main numerical results and conclusion.
This section comprises two components, one of which describe the model nonlinear stochastic heat conduction. Some theoretical results which are invoked in numerical study are listed in the other part.
Let
with Dirichlet boundary value
where
Let
Moreover, suppose
in which
This subsection is devoted to introduce basic theory related to random attractors theory as well as asymptotic behavior about system (3) which is used in this paper
For the theoretical results associated with asymptotic behavior of system (4) in deterministic case which means
In order to give the notion of global random attractors, the definition of RDS and SDS needed to be given firstly. Set
in which,
The ensuring definition of RDS was inaugurated by Arnold [1].
A RDS is continuous or differential if
The following notion of SDS can be founded in Crauel et al [4].
The SDS is not assumed to be measurable with respect to
then the SDS and RDS is coincident. However, the stochastic nonlinear heat conduction equation driven by a multiplicative white noise with a vary coefficient can generate a family of mapping satisfies the Definition 2.2, but we are not sure that it makes a cocycle. Therefore, we distinguish the aforementioned definitions here.
The coming definitions related to random attractors for RDS was established by Crauel et.al [2, 28].
𝒜( 𝒜(
Actually, Definition 2.3 is the notion of random attractor for RDS, if omit 0 in
The follow assertion provides the relationship between random attractors and invariant measures which is important to exploit the numerical results to expound the global dynamics for RDS.
The global dynamics in this paper are understood as the change in the pattern of existing probability invariant measures of SDS. With the assertion that the SDS possesses a global random attractors, based on Proposition 2.4, this investigation can be accomplished by exploit the numerical results on the structure of global random attractor. Essentially, the useful objects are global random basic attractors and global random point attractors.
With respect to the case
([18], Theorem 2.1).
The Theorem 2.5 reveals that system (3) can generate a SDS, denoted by
Utilizing Theorem 2.6, there exist global random point attractors and global random basic attractors for the SDS
([18], Theorem 5.2]).
From the Theorem 2.7, an interesting phenomenon expressed in the following manner can be derived. If min |
This Section is devoted to numerically obtain the global random attractor by stochastic subdivision algorithm method proposed in [26], subsequently, along with Proposition 2.4 analysis the global dynamics of the system. Finally, the conclusion is given.
The modal equations associated with EBS which is not display here (See Equations (A1) in 5) can be got by employing inertial manifold with delay [30] and nonlinear gakerlin method [31].
Let
The one to for order eigenvalues for 1.0270 4.1081 9.2431 16.4329
In this situation In this circumstance In this situation The assertion that In this status
By means of the aforementioned numerical results, the following affirmation on global dynamics for system (3) can be derived. When
The values of Case I Case II Case III Case IV 1 2.4 4 6 Hausdorff dimenison 0 1 2 3
Based on the global random attractors theory and Ref [18], with varying of parameter
When complex dynamics occurs, this paper expounded phenomenon simply, such as the dynamical phenomenon in Case IV. Detailed analysis should be considered by more tools. On the other hand, the global Lyapunov exponent can be invoked to describe the global dynamics of the dynamical system, which can be derived by the fact that estimation of Hausdorff dimension of global random attractors can illustrate the global dynamics of the system. However, the results on estimation of Hausdorff dimension is too conservative, therefore, we hope to study global dynamics of other mathematical physics problems by attaining the accurate global Lyapunov exponent numerically in the further.