Zero-Hopf bifurcation and Hopf bifurcation for smooth Chua’s system

Li, Junze; Liu, Yebei; Wei, Zhouchao

doi:10.1186/s13662-018-1597-8

Cited by 9 publications

(4 citation statements)

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“…Show the strange attractor for the 4-D system in 2-D projection on (x, z), (x, y), and (w, x) space respectively. Table 1 compares the proposed system (1) in terms of Lyapunov exponents to those found in the literature [15], [16], [19]. As can be observed the two positive Lyapunov exponents for the proposed system.…”

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confidence: 79%

“…Zhou et al [15] designed a smooth quadratic 4-D autonomous hyperchaotic system having complex dynamical behavior, then analyzed Hopf bifurcation, Pitchfork bifurcation, the stability of the system and other dynamical problems by applying the central manifold theory and bifurcation theory. Li et al [16] suggested the presence of zero-Hopf bifurcation by using the averaging theory and also proposed aperiodic for the proven Chua system solutions used the same theortical. Jendoubi [17] improved the presence of aperiodic solution for delayed nonautonomous non-densely partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Image scrambler based on novel 4-D hyperchaotic system and magic square with fast Walsh–Hadamard transform

2022

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A novel 4-D hyperchaotic system that have seven positive parameters in third order with thirteen terms is proposed,in this paper, the proposed chaotic behavior is proved by analysis of the Lyapunov's exponent, fractional dimension, zero-one test, sensitivity dependent on initial condition (SDIC), phase portraits, and waveform analysis, this study offers an innovative designed image encryption algorithm depending on a 4-D chaotic system using fast discrete Walsh-Hadamard transform and magic matrix that is both effective and simple for image encryption and gives it a higher level of security. This new 4-D hyperchaotic system is used to produce a random key in this algorithm. The implemented and simulated results using mathematica programs and MATLAB programs were supplied qualitatively and in figures. The proposed system is hyperchaotic, according to testing results, because it possesses two Lyapunov positive exponents.

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confidence: 79%

Section: Introductionmentioning

confidence: 99%

Image scrambler based on novel 4-D hyperchaotic system and magic square with fast Walsh–Hadamard transform

2022

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“…Bifurcation analysis is conducted for nonlinear dynamical systems to comprehend how alterations in system parameters affect the qualitative behavior of the systems [4,28,32]. There are numerous analytical and numerical methods that can be efficiently used to investigate local and global bifurcations, such as center manifold reduction, normal forms, perturbation techniques, and projection methods [7,19,21].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system

Rasul,

Salih

2024

J. Math. Computer Sci.

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This paper is devoted to investigating the local bifurcation of a chaotic jerk system. The local stability of equilibrium points is analyzed, as well as the existence of transcritical bifurcation at the origin. For the proposed jerk system, the Hopf and Zero-Hopf bifurcations are investigated at the origin. Moreover, a zero-Hopf equilibrium point at the origin is characterized for the system. By using the averaging theory of first order, a limit cycle can be bifurcated from the zero-Hopf equilibrium located at the origin. Liapunov quantities techniques are used to investigate the cyclicity of the system. It is shown that three limit cycles can be bifurcated from the origin. Finally, both self-excited chaotic attractors and hidden chaotic attractors are studied for special cases of the chaotic jerk systems using bifurcation diagrams, Lyapunov exponents, and cross-sections. All results reported in this study have been obtained using Maple software.

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“…Under the first parameter condition, the system has a periodic orbit, and under the second parameter condition, the system has five periodic orbits [15]. In 2018, Li et al considered the existence of zero-Hopf bifurcation and periodic solutions for the improved Chua system by applying the averaging theory [16]. In 2018, Salih studied the zero-Hopf bifurcation of the three-dimensional Lotka-Volterra systems [17].…”

Section: Introductionmentioning

confidence: 99%