Dynamical Analysis of the Incommensurate Fractional-Order Hopfield Neural Network System and Its Digital Circuit Realization

Abstract

Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of the incommensurate fractional-order Hopfield neural network (FOHNN) system is solved. The equilibrium point of the system is discussed, and the dissipative characteristics are verified and discussed. By varying the order values of the proposed system, different dynamical behaviors of the incommensurate FOHNN system are explored and discussed via bifurcation diagrams, the Lyapunov exponent spectrum, complexity, etc. Finally, using the DSP platform to implement the system, the results are in good agreement with those of the simulation. The actual results indicate that the system shows many complex and interesting phenomena, such as attractor coexistence and an inversion property, with dynamic changes of the order of q0, q1, and q2. These phenomena provide important insights for simulating complex neural system states in pathological conditions and provide the theoretical basis for the later study of incommensurate fractional-order neural network systems.

The Multiple Frequency Conversion Sinusoidal Chaotic Neural Network and Its Application

“…Among the basic parameters, α represents the weight of the feedback signal received by the current neuron from other neurons, which is essentially used to balance the strength of chaotic dynamics and gradient descent [1][2][3][4]. z(0) and β mainly determine the initial value and decay rate of the self-feedback term (chaos term), respectively [1][2][3][4][5][6][7][8]. For the adjustment mechanism of α, z(0), and β, it is usually expected that the early stage is dominated by chaos search (improving the global search ability, avoiding local optimization), and the late stage is dominated by gradient convergence (improving the convergence speed).…”

“…Appropriate parameter selection of the model determines whether the algorithm can find the optimal solution quickly and accurately, which is always the difficulty faced by the TCNN class optimization model. Therefore, in order to facilitate rapid, effective and clear research and the use of appropriate parameter settings, this paper summarizes all parameter settings and selection guidance by referring to many literatures [1][2][3][4][5][6][7][8][9][10][17][18][19] and the above experimental analysis and verification, as shown in Table 2 To obtain good optimization performance of the model, it is necessary to properly select and balance the relationship between the basic parameters of the model and the MFCS parameter settings. The higher the complexity of the optimization problem, the stronger the non-monotony of MFCS is required.…”

“…Appropriate parameter selection of the model determines whether the algorithm can find the optimal solution quickly and accurately, which is always the difficulty faced by the TCNN class optimization model. Therefore, in order to facilitate rapid, effective and clear research and the use of appropriate parameter settings, this paper summarizes all parameter settings and selection guidance by referring to many literatures [1][2][3][4][5][6][7][8][9][10][17][18][19] and the above experimental analysis and verification, as shown in Table 2.…”

“…Annealing factor [0, 1] (0, 1) [0.001, 0.05] z(0) [6,7] Initial value of self feedback [0, 1] [0.7, 1] [0.6, 0.8] I 0 [7,8] Positive parameter (0, ∞) [0, 1] [0.45, 0.65] ε 0 [9,10] Steepness parameter [0, ∞) [0.01, 0.5] [0.01, 0.1] A n (0) [17] Initial value of the amplitude [0, ∞) (0, 1.2] [0.1, 0.8] ε n (0) [18] Initial value of teepness (0, ∞) [0.0044, 0.32] [0.01, 0.1] a n [18] Positive parameter (0, ∞) [19] Positive parameter (0, ∞) [0.56, 1.8] [0.6, 1.5] c n [19] Weighting coefficient (0, ∞)…”

“…The structure of a TCNN network is based on the traditional Hopfield neural network (HNN) by adding self-feedback connections, which makes the network show complex chaotic dynamic behavior [3][4][5]. The optimization mechanism of TCNN is the same as that of HNN, both of which adopt the gradient descent algorithm [6]. By introducing the chaotic dynamic (self-feedback) term, TCNN can make use of the ergodic property, pseudo-randomness and non-repeatability of chaos to avoid local optimization [7,8].…”