Discrete fractional logistic map and its chaos

A discrete fractional logistic map is proposed in the left Caputo discrete delta’s sense. The new model holds discrete memory. The bifurcation diagrams are given and the chaotic behaviors are numerically illustrated.     

Monotonicity of certain integrals involving gamma distributions and their applications in multiple comparisons

Barlow and Gupta (1969) and Alam (1970) studied the monotonicity of two integrals, involving gamma distributions, that arise in certain ranking and selection problems. In this paper, we shall unify their results by studying the monotonicity of two generalized versions of integrals considered by them. We will also provide applications of derived results in study of certain multiple comparison procedures.     

A new color image cryptosystem via hyperchaos synchronization

This paper proposes a novel color image cryptosystem based on synchronization of two different six-dimensional hyperchaotic systems. In the transmitter end, we apply the drive system to generate the diffusion matrices and scrambling ones, which are used to change the image pixel value and position, respectively. Thus the ciphered image is obtained. In the receiver, synchronization of two nonidentical hyperchaotic systems can be achieved by designing the appropriate controllers. The response system is employed to yield the corresponding diffusion matrices and scrambling ones using the same generation method in the encryption algorithm. Then the cipher-image can be decrypted by the decryption algorithm, which is similar to that of the encryption process but in the reversed order. The experimental results show that the presented image cryptosystem has high security and can resist noise and crop attacks.     

Multi-objective optimal design of feedback controls for dynamical systems with hybrid simple cell mapping algorithm

This paper presents a study of multi-objective optimal design of full state feedback controls. The goal of the design is to minimize several conflicting performance objective functions at the same time. The simple cell mapping method with a hybrid algorithm is used to find the multi-objective optimal design solutions. The multi-objective optimal design comes in a set of gains representing various compromises of the control system. Examples of regulation and tracking controls are presented to validate the control design.     

Permanence and global attractivity of an impulsive delay Logistic model

In this paper, we consider an impulsive delay Logistic model. First by mathematical analysis, we obtain the maximum and minimum values of solutions of the corresponding autonomous Logistic model. Then by applying the comparison theorem and constructing some suitable Lyapunov functionals, we discuss the permanence and the global attractivity of the model, based on the boundedness of solutions of the corresponding autonomous Logistic model. An example together with its numerical simulation is given to verify our main result.     

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

Research on the target coverage algorithms for 3D curved surface

To solve the target covering problems in three-dimensional space, putting forward a deployment strategies of the target points innovatively, and referencing to the differential evolution (DE) algorithm to optimize the location coordinates of the sensor nodes to realize coverage of all the target points in 3-D surface with minimal sensor nodes. Firstly, building the three-dimensional perception model of sensor nodes, and putting forward to the blind area existing in the process of the sensor nodes sensing the target points in 3-D surface innovatively, then proving the feasibility of solving the target coverage problems in 3-D surface with DE algorithm theoretically, and reflecting the fault tolerance of the algorithm.     

Some combinatorial aspects of discrete non-linear population dynamics

Motivated by issues arising in population dynamics, we consider the problem of iterating a given analytic function a number of times. We use the celebrated technique known as Carleman linearization that turns (for a certain class of functions) this problem into simply taking the power of a real number. We expand this method, showing in particular that it can be used for population models with immigration, and we also apply it to the famous logistic map. We also are able to give a number of results for the invariant density of this map, some being related to the Carleman linearization.     

Multi-multifractality,dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice

The dynamics of random sequential partitioning of a square into ever smaller mutually exclusive rectangular blocks, which we call weighted planar stochastic lattice (WPSL), is governed by infinitely many conservation laws. Recently, we have shown one of the infinitely many conservation laws can be used as multifractal measure. In this article, we show that except the conservation of total area, each of the infinitely many non-trivial conservation laws is actually a multifractal measure and hence WPSL is a multi-multifractal. We then look at the block size distribution function and find that it exhibits dynamic scaling revealing that the spatial patterns of WPSL of different sizes are statistically self-similar. Besides, we investigate how the mean area ⟨A⟩_k of blocks with k neighbours and the average number of neighbours m_k of a typical cell that neighbours a k-sided block behaves with k. These results suggest that the Lewis law, ⟨A⟩_k∝k, is obeyed for up to k ≈ 8 and the Aboav–Weaire law, km_k∝k, is violated for the entire range of k.