The stochastic stability of a gyro-pendulum system parametrically excited by a real noise is investigated by the moment Lyapunov exponent in the paper. Using the spherical polar and non-singular linear stochastic transformations and combining these with Khasminskii's method, the diffusion process and the eigenvalue problem of the moment Lyapunov exponent are obtained. Then, applying the perturbation method and Fourier cosine series expansion, we derive an infinite-order matrix whose leading eigenvalue is the second-order expansion

There are many definitions of stochastic stability, among which the

However, it is extremely difficult to obtain the analytic expression of the moment Lyapunov exponent for an actual dynamical system according to Arnold's results due to the complexity of the noise and system. So far, almost all the results about moment Lyapunov exponents were published through the approximate analytical methods. The asymptotic expansions of the moment Lyapunov exponents on a weak noise and a small value of

Gyro-pendulums are usually used to stabilise the firing of guns on warships or tanks or to navigate cars and aircraft, and the stability of the gyro-pendulum is a classical topic in dynamics and control, which made some researchers interested in it. The mean square and almost sure stability of a gyro-pendulum under random vertical support and white-noise excitation were researched through the stochastic averaging method (Namachchivaya, 1987; Asokanthan and Ariaratnam, 2000). Recently, the stochastic stability of a planner gyro-pendulum system excited by white noises has been presented (Li and Xu, 2019). However, as we know, the ideal white noise has infinite bandwidth and is difficult to achieve in practice. Therefore, we choose a bound noise to discuss the stochastic stability of a gyro-pendulum system parametrically driven by the bounded noise in this paper. In Sect. 2, the mathematical model is given. Applying a perturbation method, the eigenvalue problem of the moment Lyapunov exponent is obtained in Sect. 3 and solved via a Fourier cosine series expansion in Sect. 4. The numerical results are given in Sect. 5. In Sect. 6, the conclusions are presented.

A typical gyro-pendulum system in the vertical configuration shown in Fig. 1 is considered, and its motion equations with a stochastic excitation

A gyro-pendulum system in vertical configuration.

For the noise excitation, we introduce a bound noise

Letting

Furthermore, according to the conclusions presented (Arnold et al., 1986b),

Generally, by solving Eq. (12), the moment Lyapunov exponent can be obtained. However, it is impossible so far since the second-order operators are so complicated and

According to the first expression of Eq. (14), the zero-order perturbation equation becomes

Applying the above same method, the corresponding adjoint differential equation of Eq. (15) is written as

From Eq. (14), the differential equation of the first-order perturbation is as follows:

Thus, Eq. (23) is simplified as

In order to obtain the joint measure

According to Eq. (14), the second-order perturbation is rewritten as

Now, we solve the eigenvalue problem shown in Eq. (43). At the two boundary points

For example, as

It is not possible to solve the analytical expression of the moment Lyapunov exponent from the eigenvalue problem defined in Eq. (46), especially for the high-order matrix

Convergence of the moment Lyapunov exponent with the increase in

Curve variation in the moment Lyapunov exponent with the increase in noise parameters

Variation in the maximum Lyapunov exponent with noise parameters

Trends of the moment Lyapunov exponent with system parameters

Effect of system parameters

Influence of system parameters

In Fig. 2, the curves of the moment Lyapunov exponent

It can be seen from the analytical expressions of the elements in matrix

In addition, moment Lyapunov exponents are not only related to noise disturbance but are also affected by system parameters. The effects of the different values of damping coefficients on the moment Lyapunov exponent are shown in Fig. 5. It is obvious that the moment stability of the system is enhanced with the increase in

In this paper, the stochastic stability of a gyro-pendulum system parametrically excited by a bounded noise is investigated through the moment Lyapunov exponent. An eigenvalue problem of the moment Lyapunov exponent is constructed by applying the theory of the stochastic dynamical system. Then, a perturbation method and Fourier cosine series expansion are used to obtain the infinite-order matrix whose leading eigenvalue is just the second-order expansion of the moment Lyapunov exponent. Furthermore, the convergence of the infinite eigenvalue sequence is numerically verified by two typical cases. Finally, the effects of system and noise parameters on the moment Lyapunov exponent are discussed. The impacts of two noise parameters on moment stability are the opposite of each other: the increase in

The code used during the study can be made available in parts or in its entirety by the corresponding author upon request.

The data used during the study can be made available in parts or in their entirety by the corresponding author upon request.

SL developed the research idea and performed the analysis and simulations. JL discussed the results and prepared the paper. All the authors provided input on the paper for revision before submission.

The contact author has declared that neither of the authors has any competing interests.

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The authors acknowledge the financial support of the National Natural Science Foundation of China and the Jiangsu Government Scholarship for Overseas Studies.

This research has been jointly supported by the National Natural Science Foundation of China (grant no. 11602098) and Jiangsu Government Scholarship for Overseas Studies (grant no. JS-2019-231).

This paper was edited by Daniel Condurache and reviewed by two anonymous referees.