Response of Quasi-Integrable and Non-Resonant Hamiltonian Systems to Fractional Gaussian Noise

The main difficulty of analyzing the response of nonlinear dynamical systems to fractional Gaussian noise (fGn) is the non-Markov property and non-usefulness of diffusion process theory. Currently, only numerical simulation can be applied to obtain the response of nonlinear systems to fGn. In the present paper, noting the rather flat property of the fGn power spectral density (PSD) in most part of frequency band, the stochastic averaging method for quasi-integrable Hamiltonian systems under wide-band noise excitation is applied to predict the response of quasi-integrable and non-resonant Hamiltonian systems to fGn. By using this method, the averaged Itô stochastic differential equations (SDEs) are established and the probability density function (PDF) of system response can be obtained from solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. All of the statistics of system response are then obtained from the PDF analytically and verified through the comparison with the simulation results.


I. INTRODUCTION
In the past half century, great achievements on nonlinear stochastic dynamics have been made [1]- [5], mainly due to the wide applications of the diffusion process theory. Recently, the fractional Gaussian noise (fGn) has been introduced to stochastic dynamics. FGn is a proper mathematical model of some real noises with long-range (or long-memory), strongly spatial and/or temporal correlation, and has been used in finance [6], signal processing [7], communication network [8] and turbulence [9], etc. It is very difficult to predict the response of nonlinear dynamical systems to fGn due to non-Markov property and non-usefulness of diffusion process theory. The exact solutions of dynamical systems excited by fGn have been obtained only for one degree-of-freedom (DOF) linear oscillator [10]. As for multi DOF linear systems and nonlinear systems excited by fGn, some approximate methods, such as the stochastic averaging method, have been developed recently.
The associate editor coordinating the review of this manuscript and approving it for publication was Qiangqiang Yuan .
The stochastic averaging methods, including the stochastic averaging methods for quasi-Hamiltonian systems, are the powerful approximate analytical methods that have been widely used in nonlinear stochastic dynamics [4], [11], [12]. Recently, the stochastic averaging method for quasi-Hamiltonian systems excited by fGn [13], [14] has been developed based on averaging principal [15], [16]. The dimension of the averaged fractional stochastic differential equation (FSDE) is less than that of original system while the dynamical characteristics of averaged system keep the same as the original system. However, due to the non-Markov property of the system response, numerical simulation has to be used for obtaining the response statistics. To develop an approximate analytical method for predicting the response of nonlinear dynamical systems to fGn then becomes an interesting topic, and that is the motivation of the present paper.
In the present paper, the power spectral density (PSD) of fGn are firstly introduced. It is pointed out that the PSD of fGn varies slowly as frequency changes in most frequency band, which indicates fGn is a wide-band process in this frequency band. Based on this observation, the stochastic averaging method for quasi-integrable Hamiltonian systems excited by wide-band noise is applied to study the response of quasi-integrable Hamiltonian systems under fGn excitation. After stochastic averaging, the response of the system is approximated as diffusion process and the transition probability density of the system is governed by an averaged Fokker-Planck-Kolmogorov (FPK) equation. All statistics of system response are then obtained from the solution of FPK equation. Finally, the obtained analytical results are verified by comparison with the results from the simulations of original and averaged FSDEs.

II. POWER SPECTRAL DENGSITY OF FGN
Similar to Gaussian white noise, fGn W H (t) is the formal derivative of fractional Brown motion (fBm) B H (t), which can be written as Mandelbrot and van Ness [17] have given the definition of fBmB H (t), an integral expansion of classical Brownian motion (Bm) B(t), as follows where parameter H is called Hurst index and 0 ≤ H ≤ 1.
The coefficient C H in Eq.
(2) reads [10] It can be seen from Eq. (2) that the fBm is more general than Bm and will reduce to the standard Bm in the case of H = 1/2. There are some useful properties of a unit fBm: The fGn is a stationary Gaussian process with following auto-correlation function (ACF) [18] Note that when H = 1/2, R(τ ) reduces to Dirac function δ(τ ), denoting fGn for H = 1/2 is a Gaussian white noise. It is pointed out that the fGn with 0 ≤ H < 1/2 is not proper for modeling physical noise since its ACF is negative and there is no PSD in the sense of traditional definition. In the present paper, only 1/2 ≤ H < 1 is considered. In the case of 1/2 ≤ H < 1, we can obtain the following PSD for fGn from Eq. (5) according to the Wiener-Khintchine relation where (·) is the gamma function. Note that when H = 1/2, the PSD S(ω) reduces to 1/2π , constant PSD of Gaussian white noise. Based on Eq. (6), the PSD of fGn for several Hurst indexes H are shown in Fig. 1. We can see from Fig. 1 that the PSD for larger frequencies, e.g. larger than 0.6, varies very slowly or flat as frequency changes. And changing H hardly change this flat feature of fGn's PSD. Thus, in this frequency band, fGn can be approximately regarded as a wide-band noise. And the stochastic averaging method for quasi-integrable Hamiltonian systems under wide-band noise excitation [19] can be applied to predict the response of dynamic systems driven by fGn.

III. PROBLEM FORMULATION AND THE MAIN METHOD
The equations of a quasi-Hamiltonian system excited by fGn is of the forṁ i, j = 1, 2, · · · , n; k = 1, · · · , 2, m. (7) where Q i and P i are generalized displacements and momenta, respectively; ε is a small parameter; Assume system (7) is quasi-integrable, then Hamiltonian of system (7) is where With certain conditions [19], system (7) has following randomly periodic solution and where A, , , are all random processes, A i are amplitude of displacements, V i (A i , i ) are the instantaneous frequency of the oscillators. Regard Eq. (10) as a transformation from (7) is transformed intȯ where By substituting A and B into Eq. (10), the h i in Eq. (13) can be written as the following form In this paper, fGn W H (t) is approximately regarded as stationary wide band process. In non-resonant case, according to Stratonovich-Khasminskii limit theorem, the processes A i (t) in Eq. (12) converge weakly to an n-dimensional Markov diffusion processes as ε → 0 in a finite time interval and is described by following averaged Itô SDEs where Usually it is more useful to convert Eq. (15) into that for H i . By using Itô differential rule and noting H i = U i (A i + B i ), the equations for H i are dH i =m i (H)dt +σ ik (H)dB k (t), i = 1, 2, · · · , n; k = 1, 2, · · · , m (17) wherē The FPK equation associated with Eq. (17) is where p = p(H, t|H 0 ) is the transition probability density of energy vector process H(t). The initial condition of Eq. (19) is For quasi-integrable and non-resonant Hamiltonian system, the stationary probability density can be calculated as [4] p Thus, the probability density for generalized displacements and generalized momenta of response of quasi-integral and non-resonant Hamiltonian system excited by fGn is obtained analytically. The following example is carrying out to show the effective of proposed method and will be compared with numerical method in the Ref. [14].
Letting X 1 = q 1 ,Ẋ 1 = p 1 , X 2 = q 2 ,Ẋ 2 = p 2 , the system (22) can be transformed to the form of quasi-Hamiltonian system (7). The associated Hamiltonian is   Using the method in Ref. [19], the following averaged FSDEs governing Hamiltonian H 1 (t), H 2 (t) can be obtained where Then, treating fGn as wide-band noise and applying the method described in the previous section, the following averaged Itô SDEs are obtained where where c i,n is the nth order coefficient in the Fourier expansions of the instantaneous frequency V i in Eq. (11) andω i = c i,0 . In terms of the following relations between A i and H i the averaged Itô SDEs for H i are then  wherem Thus, the stationary FPK equation associated with Eq. (29) is The boundary condition is  We can get the stationary PDF p(H 1 , H 2 ) by solving Eq. (31) using Peaceman-Rachford scheme with conditions (32). And the stationary PDF p(q 1 , q 2 , p 1 , p 2 ) is then obtained as follows [20] p(q 1 , q 2 , p 1 , p 2 ) =ω 1ω2 4π 2 p(H 1 , H 2 ) p(q 1 , q 2 , p 1 , p 2 )dp 1 dp 2 dq 2 , Summary, two kinds of averaged SDEs have been built to govern the Hamiltonian processes H 1 (t), H 2 (t). One is FSDEs in Eq. (24) that is obtained by using the stochastic averaging method for quasi-Hamiltonian systems excited by fGn [13], [14]. Another is Itô SDEs in Eq. (29) and its corresponding formula (33)  The results shown in Figs. 4-8 further indicate the validity of analytical expressions (33), (34) in wide range of Hurst index H provided the natural frequencies ω 1 , ω 2 of the system are larger than some values, e.g., large than 0.6, since 0.6 is roughly a boundary value between steep range and flat range of PSD (see Fig. 1). Accordingly, Fig. 6 show that the response prediction is fairly effective in flat range while ineffective in steep range. The reason, from the physical view, the response amplitude mainly depends on the system bandwidth and together with those part of noise bandwidth that fall into the range of system bandwidth.

V. CONCLUDING REMARKS
The stochastic averaging method for quasi-Hamiltonian systems under fGn excitation previously developed by us can reduce the dimension and simplify the system equation. However, the derived averaged FSDEs can only be simulated to obtain the numerical results of the response due to its non-Markov property. In the present paper, based on the observation that the PSD of fGn is quite flat for large frequency, the stochastic averaging method for quasi-integrable Hamiltonian systems under wide-band random excitation is applied to quasi-integrable and non-resonant Hamiltonian systems under fGn excitation. The response of averaged system is an approximate diffusion process and averaged FPK equation can be established and solved to yield the approximately analytical PDF and statistics of the response of original system. It has been shown via an example that the stochastic averaging method for quasi-integrable Hamiltonian systems under wide-band random excitation is applicable to quasi-integrable and nonresonant Hamiltonian systems under fGn excitation for wide range of Hurst index provided the natural frequencies are larger than some value, e.g., larger than 0.6 for studied example.
The novelty and main contribution of this paper are getting an analytical prediction for the response of quasi Hamiltonian system driven by fGn by applying the stochastic averaging method for quasi Hamiltonian system under wide-band noise for the first time. This application provide some effective parameter conditions, and more important, avoid handling the non-Markov problem of system response.