Global dynamics of the stochastic Rabinovich system

Liu, Aimin; Li, Lijie

doi:10.1007/s11071-015-2131-0

Cited by 8 publications

(5 citation statements)

References 12 publications

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“…Consider the linear equation dx = −axdt, dy = −ydt, dz = −zdt. (27) Let V (X) = 1 2 (x 2 + y 2 + z 2 ). Calculate dV (X)| (27) = (−ax 2 − y 2 − z 2 )dt ≤ −2 min{a, 1}V (X)dt.…”

Section: Random Attractor and Bifurcationmentioning

confidence: 99%

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Attractor and bifurcation of forced Lorenz-84 system

Liu

Wei

et al. 2019

Int. J. Geom. Methods Mod. Phys.

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In this paper, global dynamics of forced Lorenz-84 system are discussed, and some new results are presented. First of all, the periodic attractor is analyzed for the almost periodic Lorenz-84 system with almost periodically forcing, including the existence and the boundedness of those almost periodic solutions, and the bifurcation phenomenon in the driven system. Then, the random attractor set and the bifurcation phenomenon for the driven Lorenz-84 system with stochastic forcing are studied, including the globally exponentially attractive set, positive invariant set, random attractor and stochastic bifurcation behavior. Last but not least, some numerical simulations are also presented for verifying obtained theoretical results.

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Section: Random Attractor and Bifurcationmentioning

confidence: 99%

“…However, random noise is ubiquitous and unavoidable in the time or frequency domain. Random noise has a great impact on the dynamics of the system [17][18][19][20][21][22][23][24][25][26][27][28]. On the one hand, Lorenz [5] once pointed out that F and G should be allowed to vary periodically during a year.…”

Section: Introductionmentioning

confidence: 99%

Attractor and bifurcation of forced Lorenz-84 system

Liu

Wei

et al. 2019

Int. J. Geom. Methods Mod. Phys.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Global Dynamics of the Chaotic Disk Dynamo System Driven by Noise

Feng

Liu

et al. 2020

Complexity

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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.

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Section: Introductionmentioning

confidence: 99%

“…Quantities h, ν 1 and ν 2 are the parameters of the system: the value of h is proportional to the pump field, whereas ν 1 and ν 2 are the normalized damping decrements in the parametrically excited waves k and κ, respectively. After the original investigation in [8], the studies of this system have further been continued in [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], see also the contribution by S. Kuznetsov in this volume [25]. Although initial numerical simulations have revealed the presence of a Lorenz-like chaotic behavior in the Rabinovich system, the exact boundaries of static, periodic and chaotic dynamics in the parametric space have not been identified.…”

mentioning

confidence: 99%

Unraveling the Chaos-Land and Its Organization in the Rabinovich System

Pusuluri

Pikovsky

Shilnikov

2017

Advances in Dynamics, Patterns, Cognition

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A suite of analytical and computational techniques based on symbolic representations of simple and complex dynamics, is further developed and employed to unravel the global organization of bi-parametric structures that underlie the emergence of chaos in a simplified resonantly coupled wave triplet system, known as the Rabinovich system. Bi-parametric scans reveal the stunning intricacy and intramural connections between homoclinic and heteroclinic connections, and codimension-2 Bykov T-points and saddle structures, which are the prime organizing centers of complexity of the bifurcation unfolding of the given system. This suite includes Deterministic Chaos Prospector (DCP) to sweep and effectively identify regions of simple (Morse-Smale) and chaotic structurally unstable dynamics in the system. Our analysis provides striking new insights into the complex behaviors exhibited by this and similar systems.

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Global dynamics of the stochastic Rabinovich system

Cited by 8 publications

References 12 publications

Attractor and bifurcation of forced Lorenz-84 system

Attractor and bifurcation of forced Lorenz-84 system

Global Dynamics of the Chaotic Disk Dynamo System Driven by Noise

Unraveling the Chaos-Land and Its Organization in the Rabinovich System

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