Attractor and bifurcation of forced Lorenz-84 system

Liu, Yongjian; Wei, Zhouchao; Li, Chunbiao; Liu, Aimin; Li, Lijie

doi:10.1142/s0219887819500026

Cited by 8 publications

(4 citation statements)

References 37 publications

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“…In this situation, since the considered equation is time-invariant, we can take t 0 = 0. We recall now some standard comparison functions which are used in stability theory to characterize the stability properties and uniform asymptotic stability (see [11,16]): K is the class of functions R + → R + which are zero at the origin, strictly increasing and continuous. K ∞ is the subset of K functions that are unbounded.…”

Section: Problem Formulationmentioning

confidence: 99%

“…Stability of a system can be investigated via the first linearization method, but in general and the most powerful technique is the second direct method. For this method one usually assumes the existence of the so called Lyapunov function which is a positive definite function with negative derivative along the trajectories of the system motivated by some earlier works (see [4,9,16,[18][19][20]). Another important problem is to estimate the region of attraction around the equilibrium, that is, the problem of finding a set which contains the origin such that the limit of every trajectory starting inside is the equilibrium point.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Archives of Control Sciences

Hammami

Rettab

2020

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Many nonlinear dynamical systems can present a challenge for the stability analysis in particular the estimation of the region of attraction of an equilibrium point. The usual method is based on Lyapunov techniques. For the validity of the analysis it should be supposed that the initial conditions lie in the domain of attraction. In this paper, we investigate such problem for a class of dynamical systems where the origin is not necessarily an equilibrium point. In this case, a small compact neighborhood of the origin can be estimated as an attractor for the system. We give a method to estimate the basin of attraction based on the construction of a suitable Lyapunov function. Furthermore, an application to Lorenz system is given to verify the effectiveness of the proposed method.

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Section: Problem Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Archives of Control Sciences

Hammami

Rettab

2020

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“…The literature 50 shows the chaotic behavior of the time‐periodically perturbed Lotka–Volterra system. Liu et al 51 studied the dynamical behavior of Lorenz‐84 system with periodic disturbances; it is found that the system has quasi‐periodic attractors. Most of these scholars combine numerical simulations to further analyze the complex dynamical behaviors of differential system.…”

Section: Introductionmentioning

confidence: 99%

Qualitative geometric analysis of traveling wave solutions of the modified equal width Burgers equation

Liu

Chen

Huang

et al. 2022

Math Methods in App Sciences

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This paper devotes to the qualitative geometric analysis of the traveling wave solutions of MEW-Burgers wave equation. Firstly, MEW-Burgers equation is transformed into an equivalent planar dynamical system by using traveling wave transformation. Then the global structure of the planar system is presented, and solitary waves, kink waves (anti-kink waves), and periodic waves are found. Secondly, Jacobi stability for the planar system is studied based on KCC theory, and Jacobi stability of any point on the trajectory of system is dicussed. The dynamical behavior of the deviation vector near the equilibrium points is analyzed, and the numerical simulation is consistent with the theoretical analysis. Lyapunov stability and Jacobi stability of the equilibrium points are also compared and analyzed. The obtaining results show that Lyapunov stability of the equilibrium points of the system is not exactly consistent with Jacobi stability. Finally, the planar system with periodic disturbance is transformed into a six dimensional nonlinear system, and the periodic, quasi-periodic, and chaotic dynamical behaviors of the system are numerically simulated.

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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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Attractor and bifurcation of forced Lorenz-84 system

Cited by 8 publications

References 37 publications

Archives of Control Sciences

Archives of Control Sciences

Qualitative geometric analysis of traveling wave solutions of the modified equal width Burgers equation

Global Dynamics of the Chaotic Disk Dynamo System Driven by Noise

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