Chaos and rigorous verification of horseshoes in a class of Hopfield neural networks

Abstract

In this paper, chaos in a new class of threedimensional continuous time Hopfield neural networks is investigated. Numerical experiments show that this class of Hopfield neural networks can have chaotic attractors and limit cycles for different parameter configurations. By virtue of horseshoes theory in dynamic systems, rigorous computerassisted verifications are done for their chaotic behavior. In terms of topological entropy, quantitative interpretations of these neural networks' complexity are given. A brief analysis is also presented about their robustness.

Novel compressive sensing image encryption using the dynamics of an adjustable gradient Hopfield neural network

“…The studies carried out on AHNNs have the advantage that they would allow an understanding of a certain number of pathologies affecting memory function and provide a starting point for their resolution. As a result, dynamic behaviors such as chaos [1][2][3], hyperchaos [4][5][6], multistability [7][8][9], and hidden attractors [4,5] have piqued the interest of researchers.…”

“…To facilitate theoretical analysis and experimental implementation, many small HNNs have been intensively studied [1,3,5,[17][18][19][20][21][22]. Complex dynamic behaviors such as periodic, chaotic, or coexisting attractors, crisis scenarios [1,3,5,[17][18][19][20]22], hyperchaos [5,6], relaxation oscillations [21], and hidden attractors [5], to name a few, have generally been revealed in these small HNNs.…”

“…To facilitate theoretical analysis and experimental implementation, many small HNNs have been intensively studied [1,3,5,[17][18][19][20][21][22]. Complex dynamic behaviors such as periodic, chaotic, or coexisting attractors, crisis scenarios [1,3,5,[17][18][19][20]22], hyperchaos [5,6], relaxation oscillations [21], and hidden attractors [5], to name a few, have generally been revealed in these small HNNs. Most of these small HNNs listed are at least three-order autonomous [1,3,5,[17][18][19][20], but there are still other smaller non-autonomous and autonomous second-order ones that have been recently developed.…”

“…Complex dynamic behaviors such as periodic, chaotic, or coexisting attractors, crisis scenarios [1,3,5,[17][18][19][20]22], hyperchaos [5,6], relaxation oscillations [21], and hidden attractors [5], to name a few, have generally been revealed in these small HNNs. Most of these small HNNs listed are at least three-order autonomous [1,3,5,[17][18][19][20], but there are still other smaller non-autonomous and autonomous second-order ones that have been recently developed. As such, we can list the works of [23][24][25][26] who studied in-depth HNNs models with two neurons and highlighted complex dynamic behaviors.…”

Zhang neural network and its application to Newton iteration for matrix square root estimation

“…Being another important type of solution, many parallel-processing computational schemes, including various dynamic recurrent neural networks (RNN) solvers, have been developed, analyzed, and implemented on specific architectures [9][10][11][12][13]. The RNN (or termed neural dynamics) approach is now regarded as a powerful alternative to online computation because of its parallel distributed nature and, more importantly, the suitability of hardware realization [11][12][13][14][15][16].…”