Bifurcation and Chaos in Noninteger Order Cellular Neural Networks

Abstract

In this paper a new class of Cellular Neural Networks (CNNs) is introduced. The peculiarity of the new CNN model consists in replacing the traditional first order cell with a noninteger order one. The introduction of fractional order cells, with a suitable choice of the coupling parameters, leads to the onset of chaos in a two-cell system of a total order of less than three. A theoretical approach, based on the interaction between equilibrium points and limit cycles, is used to discover chaotic motions in fractional CNNs.

Adaptive Synchronization for a Class of Uncertain Fractional-Order Neural Networks

“…Remark 1. Assumption 1 can be seen in many literatures [3,13,[15][16][17][18][19][22][23][24]. Besides, different from most of above works, in this paper, we assume that the Lipschitz constant L i is unknown.…”

“…Consequently, many researchers utilize fractional-order calculus in neural networks to construct fractional-order neural networks, which can be more appropriate to describe the dynamical behavior of the neurons in neural networks, such as "memory". In [11], Arena et al first give a cellular neural network with fractional-order cells. In [12], Petráč introduces a fractional-order 3-cell network which exhibits limit cycles and stable orbits for different parameter values.…”

“…It is known that there are many synchronization results about integer-order neural networks (see, [16,17] and references therein). Since bifurcations and chaos of fractional-order neural networks have been firstly appeared in [18,19], some significant results on fractional-order neural networks have been given. In [20], a fractional-order Hopfield neural model is discussed, and its stability is studied by energy-like functions.…”

Stability Analysis and Synchronization for a Class of Fractional-Order Neural Networks

“…For example, bifurcations and chaos phenomenon in FONNs are studied in [12][13][14]. In [15], energy-like functions are utilized to study the stability of FONNs.…”

Chaotic convective behavior and stability analysis of a fractional viscoelastic fluids model in porous media

“…Hartley et al introduced the fractional-order Chua's system and showed that with the order less than 3, the chaotic behavior is still possible [20]. In [21], the fractional-order cellular neural network was considered, and Ibrahima N'Doye and Taous-Meriem Laleg-Kirati are with Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST). Thuwal 23955-6900, KSA.…”