Multistability and Formation of Spiral Waves in a Fractional-Order Memristor-Based Hyperchaotic Lü System with No Equilibrium Points

Multistablity analysis and formation of spiral wave in the fractional-order nonlinear systems is a recent hot topic. In this paper, dynamics, coexisting attractors, complexity, and synchronization of the fractional-order memristor-based hyperchaotic Lü system are investigated numerically by means of bifurcation diagram, Lyapunov exponents (LEs), chaos diagram, and sample entropy (SampEn) algorithm. The results show that the system has rich dynamics and high complexity. Meanwhile, coexisting attractors in the system are observed and hidden dynamics are illustrated by changing the initial conditions. Finally, the network based on the system is built, and the emergence of spiral waves is investigated and chimera states are observed.

A New Chaos-Based Color Image Encryption Scheme with an Efficient Substitution Keystream Generation Strategy

This paper suggests a new chaos-based color image cipher with an efficient substitution keystream generation strategy. The hyperchaotic Lü system and logistic map are employed to generate the permutation and substitution keystream sequences for image data scrambling and mixing. In the permutation stage, the positions of colored subpixels in the input image are scrambled using a pixel-swapping mechanism, which avoids two main problems encountered when using the discretized version of area-preserving chaotic maps. In the substitution stage, we introduce an efficient keystream generation method that can extract three keystream elements from the current state of the iterative logistic map. Compared with conventional method, the total number of iterations is reduced by 3 times. To ensure the robustness of the proposed scheme against chosen-plaintext attack, the current state of the logistic map is perturbed during each iteration and the disturbance value is determined by plain-pixel values. The mechanism of associating the keystream sequence with plain-image also helps accelerate the diffusion process and increase the degree of randomness of the keystream sequence. Experimental results demonstrate that the proposed scheme has a satisfactory level of security and outperforms the conventional schemes in terms of computational efficiency.

Function Projective Dual Synchronization with Uncertain Parameters of Hyperchaotic Systems

Function projective dual synchronization between two pairs of hyperchaotic systems with fully unknown parameters for drive and response systems is investigated. On the basis of the Lyapunov stability theory, a suitable and effective adaptive control law and parameters update rule for unknown parameters are designed, such that function projective dual synchronization between the hyperchaotic Chen system and the hyperchaotic Lü system with unknown parameters is achieved. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed method.

Controlling Hopf Bifurcation of a New Modified Hyperchaotic Lü System

Controlling Hopf bifurcation of a new modified hyperchaotic Lü system is investigated in this paper. A hybrid control strategy using both state feedback and parameter control is proposed. The control strategy realizes the delay of Hopf bifurcation. Furthermore, by applying the normal form theory, the stability of the bifurcation is determined. Numerical simulation results are given to support the theoretical analysis.

The Multistage Homotopy Perturbation Method for Solving Chaotic and Hyperchaotic Lü System

The multistage homotopy-perturbation method (MHPM) is applied to the nonlinear chaotic and hyperchaotic Lü systems. MHPM is a technique adapted from the standard homotopy-perturbation method (HPM) where the HPM is treated as an algorithm in a sequence of time intervals. To ensure the precision of the technique applied in this work, the results are compared with a fourth-order Runge-Kutta method and the standard HPM. The results show that the MHPM is an efficient and powerful technique in solving both chaotic and hyperchaotic systems.

On the piecewise-spectral homotopy analysis method and its convergence: solution of hyperchaotic Lü system

Abstract- In this paper, a novel modification of the spectral-homotopy analysis method (SHAM) technique for solving highly nonlinear initial value problems that model systems with chaotic and hyper-chaotic behaviour is presented. The proposed method is based on implementing the SHAM on a sequence of multiple intervals thereby increasing it’s radius of convergence to yield highly accuratemethod which is referred to as the piece-wise spectral homotopy analysis method (PSHAM). We investigate the application of the PSHAM to the L¨u system [20] which is well known to display periodic, chaotic and hyper-chaotic behaviour under carefully selected values of it’s governing parameters. Existence and uniqueness of solution of SHAM that give a guarantee of convergence of SHAM, has been discussed in details. Comparisons are made between PSHAMgenerated results and results from literature and Runge-Kutta generated results and good agreement is observed.