Abstract
The 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. The 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.
1. Introduction
The 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. The behaviour of the system is far different from that of the single disk dynamo, which never has a reversal of the electric current. Then, a simple mechanical model used to study the reversals of the Earth’s magnetic field is a two-disc dynamo system idealized by Rikitake. The 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 x3 and x4 and the currents generated by x1 and x2, respectively. Then, with appropriate normalization of variables, the dynamical equations can be described by the following set of ordinary differential equations [3, 4]:where q1 and q2 are the torques applied to the rotors and μ1, μ2, ϵ1, and ϵ2 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 represent the power consumption and mechanical damping dissipation of disk dynamo, respectively. When the parameters are μ1 = 3, μ2 = 1, ϵ1 = 0.1, ϵ2 = 0.2, q1 = 3, and q2 = 1 for initial states (x1(0), x2(0), x3(0), x4(0)) = (2.2, 2.0, 10.5, 20), the numerical simulation shows that the corresponding Lyapunov exponents are 0.28, 0, −0.10, and −4.47. There exists one positive Lyapunov exponent suggest that system (1) has a chaotic attractor. The chaotic attractor’s projections in the coordinate planes x1 − x2 − x3 and x2 − x3 − x4 are shown in Figure 1.
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(b)
One the one hand, since the Lorenz system [5] was presented, there is a huge volume of the literature devoted to the studies of the Lorenz system and other classical chaotic systems, which are closely related but not topologically equivalent to the Lorenz system, such as Chen system [6], Lü system [7], and Yang system [8]. In a sense defined by Vaněček and Čelikovský [9, 10], the Chen system is a dual system to the Lorenz system and the Lü system and Yang system represent a transition between the Lorenz and the Chen systems. For the Lorenz family system, mathematicians, physicists, and engineers from various fields have studied the characteristics of systems, bifurcations, routes to chaos, essence of chaos, and chaos synchronization. By ignoring mechanical damping dissipation that parameters ϵ1 = ϵ2 = 0 and setting q1 = q2 = 1 and μ1 = μ2 = μ, we can write x3 = z and x4 = z − α, where α is a constant of the motion. Finally, coupled dynamos (1) can be written in the following simple form [11]:
System (2) has a three-dimensional attractor similar to the Lorenz attractor although both systems are obviously not topologically equivalent [11]. The 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, [11–16] 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 well-known 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 μ1, μ2, ϵ1, ϵ2, q1, and q2 by μ1 ⟶ μ1 + σ1dW(t), μ2 ⟶ μ2 + σ2dW(t), ϵ1 ⟶ ϵ1 + σ3dW(t), ϵ2 ⟶ ϵ2 + σ4dW(t), q100 ⟶ q1 + q10dW(t), and q2 ⟶ q2 + q20dW(t), where are the mutually independent Brownian motions. Then, 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. The corresponding stochastic case uses the same parameters and initial values. Let μ1 = 3, μ2 = 1, ϵ1 = 0.1, ϵ2 = 0.2, q1 = 3, and q2 = 1 and initial states (x1(0), x2(0), x3(0), x4(0)) = (2.2, 2.0, 10.5, 20) additionally have the perturbed parameters σ1 = 0.1, σ2 = 0.1, σ3 = 0.01, σ4 = 0.01, q10 = 0.1, and q20 = 0.1. The projections in the coordinate planes x1 − x2 − x3 and x2 − x3 − x4 are shown in Figure 2. Comparing Figures 1 and 2, we can see the difference between the deterministic case and stochastic case. Actually, the behavior of system will change even if the parameters suffer small perturbation. Suppose that other parameters remain unchanged and only ϵ2 change; let μ1 = 3, μ2 = 1, ϵ1 = 0.1, ϵ2 = 0.2, q1 = 3, and q2 = 1, initial states (x1(0), x2(0), x3(0), x4(0)) = (2.2, 2.0, 10.5, 20), and the perturbed parameters σ1 = 0, σ2 = 0, σ3 = 0, σ4 = 0.01, q10 = 0, and q20 = 0. Time series x1 diagram of the deterministic case and stochastic case are shown in Figure 3.
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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. This 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 [22–26]. 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. The 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.
This 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. The conclusions are given in Section 5.
2. Cylindrical Bound
Theorem 1. Let and . Suppose that the parameters , , , and , σ1 ≥ 0(i = 1, 2, 3, 4). Then, the set Ω is the bound for system (3), in the sense that system (3) is the cylindrical bound, where
Proof.
Step 1. Construct a positive definite and radically unbounded Lyapunov function on :Applying Itô’s formula, one haswhereSimilar to the proof of Theorem 1, we can obtainTherefore, one has ; that is to say, the following inequality holds as t ⟶ +∞:
Step 2. Construct a positive definite and radically unbounded Lyapunov function on :Similar to the proof of Step 1, we can obtainwhereTherefore, the following inequality holds as t ⟶ +∞:By Step 1 and Step 2, system (3) is the cylindrical bound.
Remark 1. Let σ1 = σ2 = σ3 = σ4 = 0; then system (3) is deterministic. Theorem 2 contains the results given in [12] as special cases.
Let μ1 = 3, μ2 = 1, ϵ1 = 0.1, ϵ2 = 0.2, q1 = 3, q2 = 1, σ1 = σ2 = 0.1, σ3 = σ4 = 0.01, and q10 = q20 = 0.1, and initial values (x0, y0, z0, u0) = (2.2, 2, 2.5, 3). Calculate l13 = 0.099950 and L24 = 0.199950. We give the following estimate of the ultimate boundary:The corresponding projections of exponentially attractive sets are shown in Figure 4.
And we also have the following results:The 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.
3. Globally Exponentially Attractive Set
Theorem 2. Let , and . Suppose that the parameters , , , and . Then, for any constant λ > 0, the following estimate holds on system (3):
In particular,is a globally exponential attractive set of system (3), where
Proof. Define the Lyapunov on , whereApplying Itô’s formula to (19), one haswhereThen,From (22) and (23), we can obtainwhereFrom (24) and the calculating the expectation, one obtainsFrom above inequality, one can obtainLet . When EV(X) − L > 0, EV(X0) − L > 0, the following estimate holds:Thus,That is,
Theorem 3. Let , and . Suppose that the parameters , , , and . Then, for any constant λ > 0 and , the following estimate holds on system (3):
In particular,is a globally exponential attractive set of system (3), where
Proof. The proof is the same as that for Theorem 2; we omit it here.
Remark 2. Let σ1 = σ2 = σ3 = σ4 = q10 = q20 = 0; then, system (3) is deterministic. Theorem 3 contain the results given in [12] as special cases.
In Theorem 3, let μ1 = 3, μ2 = 1, ϵ1 = 0.1, ϵ2 = 0.2, q1 = 3, q2 = 1, σ1 = σ2 = 0.1, σ3 = σ4 = 0.01, q10 = q20 = 0.1, and initial values (x0, y0, z0, u0) = (2.2, 2, 10.5, 20). We give the following estimate of the ultimate boundary:This is the globally exponential attractive set and positive invariant set of the stochastic disk dynamo system.
Then, we have the following results of the ultimate boundary about x1 − x2 − x3, x1 − x2 − x4, x1 − x3 − x4, and x2 − x3 − x4, which are the exponentially attractive sets of the stochastic disk dynamo system:The numerical solutions, which are stochastic processes, of stochastic dynamo system (3) are obtained by the Euler–Maruyama method. The simulated time series about is displayed in Figure 5. In addition, the stochastic processes’ scopes isand the corresponding expectation isIt is nice to see that the simulation results and the theoretical results of (34) are consistent.
4. Stochastic Stability
The purpose of this section is to seek condition for the asymptotic behavior of system (3).
Theorem 4. When perturbed parameters σ3 = σ4 = q10 = q20 = 0, suppose that the parameters and . If , the equilibrium position of system (3) is stochastically asymptotically stable.
Proof. Let . Then,When the parameters , , and , the matrix Q is positive-definite. Thus, LV is negative-definite. Then, from Theorem 4.2.3 of [35], the equilibrium position of system (3) is stochastically asymptotically stable.
Remark 3. When the perturbed parameters σ1 = σ2 = σ3 = σ4 = q10 = q20 = 0, system (3) is deterministic. Suppose μ1 = μ2, ϵ1 = ϵ2, and q1 = q2; then, condition is reduced to . That is to say, the deterministic disk dynamo system is stable when . That conclusion is coincident with the result of the literature [4].
Remark 4. Comparing the conditions of Theorem 3 and Theorem 4, the results show that the asymptotically stability of the stochastic disk dynamo system occurs when , which means the stochastic disk dynamo system will not show chaotic behavior.
Let u1 = 2.5, u2 = 2.6, ϵ1 = 0.5, ϵ2 = 1, q1 = 1.5, q2 = 2, σ1 = 1, σ2 = 1, and σ3 = σ4 = q10 = q20 = 0, and initial values (x0, y0, z0, u0) = (2.2, 2, 10.5, 20). In Figure 6, the number results show that the trivial solution of system (3) is stochastically asymptotically stable.
5. Conclusions
The 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. The 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. The 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 11961074 and 11971414), Natural Science Foundation of Guangxi Province (Grant nos. 2018GXNSFDA281028, 2017GXNSFAA198234, and 2016GXNSFBA380170), the Youth Project of Hunan Provincial Education Department (Grant nos. 18B518 and 18B082), the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document no. [2018] 35), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (CUGGC05).