Global Dynamics of the Chaotic Disk Dynamo System Driven by Noise

.e disk dynamo system, which is capable of chaotic behaviours, is obtained experimentally from two disk dynamos connected together. It models the geomagnetic field and is used to explain the reversals in its polarity. Actually, the parameters of the chaotic systems exhibit random fluctuation to a greater or lesser extent, which can carefully describe the disturbance made by environmental noise. .e global dynamics of the chaotic disk dynamo system with random fluctuating parameters are concerned, and some new results are presented. Based on the generalized Lyapunov function, the globally attractive and positive invariant set is given, including a two-dimensional parabolic ultimate boundary and a four-dimensional ellipsoidal ultimate boundary. Furthermore, a set of sufficient conditions is derived for all solutions of the stochastic disk dynamo system being global convergent to the equilibrium point. Finally, numerical simulations are presented for verification.


Introduction
e magnetic field has reversed its polarity many times along geological history [1]. To geophysics, their fundamental goal is a coherent understanding of the structure and dynamics of the Earth's interior. A number of investigators worked hard in order to establish the state of the Earth's dynamo. Bullard studied a disk dynamo with the intention of discussing possible analogies between them and those of a homogeneous dynamo which is supposed to be the origin of the magnetic field of the Earth and other celestial bodies. Before long, Japanese geophysicist Rikitake [2] found that reversals of electric current generated by a circuit can often occur even in a very simple system such as the one with two disk dynamos. e behaviour of the system is far different from that of the single disk dynamo, which never has a reversal of the electric current. en, a simple mechanical model used to study the reversals of the Earth's magnetic field is a two-disc dynamo system idealized by Rikitake. e model consists of two identical single Faraday-disk dynamos of the Bullard type coupled together. For simplicity, we denote the angular velocities of their rotors by x 3 and x 4 and the currents generated by x 1 and x 2 , respectively. en, with appropriate normalization of variables, the dynamical equations can be described by the following set of ordinary differential equations [3,4]: where q 1 and q 2 are the torques applied to the rotors and μ 1 , μ 2 , ϵ 1 , and ϵ 2 are the positive constants representing dissipative effects of the disk dynamo system. Rather, from the physical meaning of the equation, the parameters μ and ϵ System (2) has a three-dimensional attractor similar to the Lorenz attractor although both systems are obviously not topologically equivalent [11]. e chaotic behavior and other properties, synchronization and control of the disk dynamo system and disk dynamo-like chaotic systems (2), were extensively studied (see, for instance, [11][12][13][14][15][16] and their references).
On the other hand, Arnold [17] has pointed out that the parameters in the chaotic systems exhibit random fluctuation to a greater or lesser extent due to various environmental noise. Scholars usually estimate them by average values plus some error terms [18]. In general, by the wellknown central limit theorem, the error terms follow normal distributions. For the best incorporate (natural) randomness into the mathematical description of the phenomena and to provide a more accurate description of it, we model the stochastic disk dynamo system by replacing the parameters μ 1 , μ 2 , ϵ 1 , ϵ 2 , q 1 , and q 2 by μ 1 ⟶ μ 1 + σ 1 dW(t), μ 2 ⟶ μ 2 + σ 2 dW(t), ϵ 1 ⟶ ϵ 1 + σ 3 dW(t), ϵ 2 ⟶ ϵ 2 + σ 4 dW(t), q 100 ⟶ q 1 + q 10 dW(t), and q 2 ⟶ q 2 + q 20 dW(t), where W(t) are the mutually independent Brownian motions.
en, one gets the following system of stochastic differential equations: To illustrate the stochastic effects clearly, we performed simulations for the corresponding stochastic case of Figure 1.
Chaos synchronization is a very important topic in chaos theory. Enormous research activities have been carried out in chaos synchronization by many researchers from different disciplines, and lots of successful experiments have been reported. Many scholars, by using capacitor coupling [19], induction coil coupling [20], and resistance coupling [21] to realize the synchronization of chaotic systems, have obtained good results. In chaotic synchronization, the boundedness of the system is a very important prerequisite. In fact, ultimate boundedness of chaotic dynamical systems is always one of the fundamental concepts in dynamical systems. is plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors and the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. Technically, to locate and estimate the relative position of the attractor is a difficult work even in a deterministic system [22][23][24][25][26]. For the deterministic system, Yu and Liao [27] give the concept of the exponential attractive set and estimate the globally attractive and positive invariant set of the typical Lorenz system. For the stochastic system, some results of the estimation global attractive set have also been obtained, for the stochastic Lorenz-Stenflo system [18], the stochastic Lorenz-Haken system [28], the stochastic Lorenz-84 system [29], the stochastic Lorenz system family [30], the stochastic Rabinovich system [31,32], and other stochastic systems [33,34].
In this paper, by using a technique combining the generalized Lyapunov function theory and optimization, globally exponential attractive set and a four-dimensional ellipsoidal ultimate bound are derived, which can help us to locate the relative position of the attractor. e two-dimensional parabolic ultimate bound is also established. And numerical results to estimate the ultimate bound are also presented for verification. We hope that the investigation of this paper can help understanding the rich dynamic of the stochastic disk dynamo system and offer some enlightenments for the study of the reversals of the Earth's magnetic field.
is paper is organized as follows. In Section 2, the cylindrical bound of stochastic disk dynamo system (3) is presented. In Section 3, globally exponential attractive set and positive invariant set of the system are derived. In Section 4, the stochastic stability of system (3) is studied. In each section, we also give corresponding numerical results, respectively. e conclusions are given in Section 5. 3,4). en, the set Ω is the bound for system (3), in the sense that system (3) is the cylindrical bound, where
And we also have the following results: e numerical solutions, which are stochastic processes, of stochastic dynamo system (3) are obtained by the Euler-Maruyama method. All the stochastic processes' scopes and the ultimate boundary of the corresponding expectations are listed in Table 1. From Table 1, we are pleased to see that the simulation results and the theoretical results of (14) and (15) are consistent.

Stochastic Stability
e purpose of this section is to seek condition for the asymptotic behavior of system (3).

Conclusions
e coupled dynamo system is a nonlinear dynamical system which is capable of chaotic behaviours. It models the geomagnetic field and is used to explain the reversals in its polarity. Actually, the parameters of the chaotic systems exhibit random fluctuation to a greater or lesser extent, which can carefully describe the disturbance made by environmental noise. e global dynamics of the chaotic disk dynamo system with random fluctuating parameters are concerned, and some new results are presented. Based on the generalized Lyapunov function, the globally attractive and positive invariant set is given, including a two-dimensional parabolic ultimate boundary and a four-dimensional ellipsoidal ultimate boundary. Furthermore, a set of sufficient conditions is derived for all solutions of the stochastic disk dynamo system being global convergent to the equilibrium point. e stochastic disk dynamo system will not show chaotic behavior when the system is stable. Finally, numerical simulations are presented for verification.

Data Availability
All data generated or analyzed during this study are included in this article.