BIFURCATION AND CHAOS ANALYSIS FOR A DELAYED TWO-NEURAL NETWORK WITH A VARIATION SLOPE RATIO IN THE ACTIVATION FUNCTION

2012 ◽  
Vol 22 (05) ◽  
pp. 1250105 ◽  
Author(s):  
ZIGEN SONG ◽  
JIAN XU
Keyword(s):  
Neural Network ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Activation Function ◽  
Phase Portraits ◽  
Stable Equilibrium Point ◽  
Slope Ratio ◽  
Chaotic Motions ◽  
Complex Phenomena ◽  
Central Manifold Reduction

In this paper, some complex phenomena of dynamical bifurcations are shown for a two-neural network with delay coupling. The sigmoid activation function with slope ratio, a monotonically increasing function, is proposed to consider the relations of the sigmoid and Hardlim functions. The equilibrium points are studied analytically in detail in terms of the characteristic equation and static bifurcation. The central manifold reduction and normal form method are employed to determine Hopf bifurcation and its stability. The stable equilibrium points and periodic motions are observed in different parameter regions. Effects of slope ratio and delayed coupling on dynamic behaviors are investigated by the numerical simulation, such as Poincare map, phase portraits, and power spectrum. Various active transitions to chaos and the corresponding critical boundaries on the focused parameter regions are obtained to classify dynamical behaviors including stable equilibrium point, periodic solution, 2-torus, 3-torus, and then the chaotic motions.

scholarly journals Research of Trajectory Optimization Approaches in Synthesized Optimal Control

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 336
Author(s):  
Askhat Diveev ◽  
Elizaveta Shmalko
Keyword(s):  
Optimal Control ◽  
Equilibrium Point ◽  
Trajectory Optimization ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Control Object ◽  
Symbolic Regression ◽  
Optimal Location ◽  
Stable Equilibrium Point ◽  
Stabilization System

This article presents a study devoted to the emerging method of synthesized optimal control. This is a new type of control based on changing the position of a stable equilibrium point. The object stabilization system forces the object to move towards the equilibrium point, and by changing its position over time, it is possible to bring the object to the desired terminal state with the optimal value of the quality criterion. The implementation of such control requires the construction of two control contours. The first contour ensures the stability of the control object relative to some point in the state space. Methods of symbolic regression are applied for numerical synthesis of a stabilization system. The second contour provides optimal control of the stable equilibrium point position. The present paper provides a study of various approaches to find the optimal location of equilibrium points. A new problem statement with the search of function for optimal location of the equilibrium points in the second stage of the synthesized optimal control approach is formulated. Symbolic regression methods of solving the stated problem are discussed. In the presented numerical example, a piece-wise linear function is applied to approximate the location of equilibrium points.

scholarly journals Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xia Huang ◽  
Zhen Wang ◽  
Yuxia Li
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Fractional Order ◽  
Hopfield Neural Network ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Chaotic Motions ◽  
Dynamical Behaviors ◽  
Nonlinear Dynamics And Chaos

A fractional-order two-neuron Hopfield neural network with delay is proposed based on the classic well-known Hopfield neural networks, and further, the complex dynamical behaviors of such a network are investigated. A great variety of interesting dynamical phenomena, including single-periodic, multiple-periodic, and chaotic motions, are found to exist. The existence of chaotic attractors is verified by the bifurcation diagram and phase portraits as well.

scholarly journals An Evolutionary Game Analysis of Internet Public Opinion Events at Universities: A Case from China

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Hongying Wen ◽  
Kairong Liang ◽  
Yiquan Li
Keyword(s):  
Public Opinion ◽  
University Students ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Evolutionary Game ◽  
Stable Equilibrium Point ◽  
University Administration ◽  
Negative Effects ◽  
Internet Medium ◽  
The University

Internet public opinion events at universities in China occurred frequently, creating painful repercussions for reputation and stability of colleges and universities. To better cope with the problem, this paper explores an evolutionary mechanism of the university Internet public opinion events. Firstly, we discuss the interactions and behavior of three key participants: an Internet medium, university students as a whole, and administration. Secondly, we construct a tripartite evolutionary game model consisting of an Internet medium, student group, and university administration and then analyze and obtain the differential dynamic equations and equilibrium points. Subsequently, the evolutionary stable equilibrium is further analyzed. Finally, we employ numerical studies to examine how the tripartite behavior choices affect evolutionary paths and evolutionary equilibrium strategies. Results are derived as follows: under certain conditions, there exists an asymptotically stable equilibrium point for the tripartite evolutionary game. On the one hand, appropriate penalties and rewards should be provided to foster objectives and fair behaviors of the network medium. On the other hand, university students should be educated and guided to deal rationally with negative effects of Internet public opinion events. Moreover, online real-name authentication is an important and necessary measure. Finally, the university administration should release truthful, timely, and comprehensive information of Internet public opinion events to mitigate potential negative impacts.

The Qualitative Analysis of Two Populations Commensalisms Model

2012 ◽  
Vol 524-527 ◽  
pp. 3705-3708
Author(s):  
Guang Cai Sun
Keyword(s):  
Qualitative Analysis ◽  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Mathematics Model ◽  
Stable Equilibrium Point ◽  
The Stability ◽  
Two Populations

This paper deals with the mathematics model of two populations Commensalisms symbiosis and the stability of all equilibrium points the system. It has given the conclusion that there is only one stable equilibrium point the system. This paper also elucidates the biology meaning of the model and its equilibrium points.

scholarly journals Author’s reply to: Comments on “Asymptotically stable equilibrium points in new chaotic systems”

Nova Scientia ◽  
2017 ◽  
Vol 9 (19) ◽  
pp. 906-909
Author(s):  
K. Casas-García ◽  
L. A. Quezada-Téllez ◽  
S. Carrillo-Moreno ◽  
J. J. Flores-Godoy ◽  
Guillermo Fernández-Anaya
Keyword(s):  
Dynamic Systems ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Linear Quadratic ◽  
The Monte Carlo Method

Since theorem 1 of (Elhadj and Sprott, 2012) is incorrect, some of the systems found in the article (Casas-García et al. 2016) may have homoclinic or heteroclinic orbits and may seem chaos in the Shilnikov sense. However, the fundamental contribution of our paper was to find ten simple, three-dimensional dynamic systems with non-linear quadratic terms that have an asymptotically stable equilibrium point and are chaotic, which was achieved. These were obtained using the Monte Carlo method applied specifically for the search of these systems.

Four-Wing Hidden Attractors with One Stable Equilibrium Point

2020 ◽  
Vol 30 (06) ◽  
pp. 2050086 ◽  
Author(s):  
Quanli Deng ◽  
Chunhua Wang ◽  
Linmao Yang
Keyword(s):  
Equilibrium Point ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Stable Node ◽  
Stable Equilibrium Point ◽  
The Novel ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
Attraction Basin

Although multiwing hidden attractor chaotic systems have attracted a lot of interest, the currently reported multiwing hidden attractor chaotic systems are either with no equilibrium point or with an infinite number of equilibrium points. The multiwing hidden attractor chaotic systems with stable equilibrium points have not been reported. This paper reports a four-wing hidden attractor chaotic system, which has only one stable node-focus equilibrium point. The novel system can also generate a hidden attractor with one-wing and hidden attractors with quasi-periodic and periodic coexistence. In addition, a self-excited attractor with one-wing can be generated by adjusting the parameters of the novel system. The hidden attractors of the novel system are verified by the cross-section of attraction basins. And the hidden behavior is investigated by choosing different initial states. Moreover, the coexisting transient four-wing phenomenon of the self-excited one-wing attractor system is studied by the time domain waveforms and attraction basin. The dynamical characteristics of the novel system are studied by Lyapunov exponents spectrum, bifurcation diagram and Poincaré map. Furthermore, the novel hidden attractor system with four-wing and one-wing are implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations.

scholarly journals Identification and Machine Learning Prediction of Nonlinear Behavior in a Robotic Arm System

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1445
Author(s):  
Cheng-Chi Wang ◽  
Yong-Quan Zhu
Keyword(s):  
Neural Network ◽  
Lyapunov Exponent ◽  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Robotic Arm ◽  
Maximum Lyapunov Exponent ◽  
Double Pendulum ◽  
Chaotic Motions ◽  
Robotic Arms

In this study, the subject of investigation was the dynamic double pendulum crank mechanism used in a robotic arm. The arm is driven by a DC motor though the crank system and connected to a fixed side with a mount that includes a single spring and damping. Robotic arms are now widely used in industry, and the requirements for accuracy are stringent. There are many factors that can cause the induction of nonlinear or asymmetric behavior and even excite chaotic motion. In this study, bifurcation diagrams were used to analyze the dynamic response, including stable symmetric orbits and periodic and chaotic motions of the system under different damping and stiffness parameters. Behavior under different parameters was analyzed and verified by phase portraits, the maximum Lyapunov exponent, and Poincaré mapping. Firstly, to distinguish instability in the system, phase portraits and Poincaré maps were used for the identification of individual images, and the maximum Lyapunov exponents were used for prediction. GoogLeNet and ResNet-50 were used for image identification, and the results were compared using a convolutional neural network (CNN). This widens the convolutional layer and expands pooling to reduce network training time and thickening of the image; this deepens the network and strengthens performance. Secondly, the maximum Lyapunov exponent was used as the key index for the indication of chaos. Gaussian process regression (GPR) and the back propagation neural network (BPNN) were used with different amounts of data to quickly predict the maximum Lyapunov exponent under different parameters. The main finding of this study was that chaotic behavior occurs in the robotic arm system and can be more efficiently identified by ResNet-50 than by GoogLeNet; this was especially true for Poincaré map diagnosis. The results of GPR and BPNN model training on the three types of data show that GPR had a smaller error value, and the GPR-21 × 21 model was similar to the BPNN-51 × 51 model in terms of error and determination coefficient, showing that GPR prediction was better than that of BPNN. The results of this study allow the formation of a highly accurate prediction and identification model system for nonlinear and chaotic motion in robotic arms.

ADAPTABILITY AND SENSITIVITY OF COMPLEX SYSTEMS

2013 ◽  
Vol 23 (09) ◽  
pp. 1350163 ◽  
Author(s):  
ZHI-CHENG YE ◽  
QING-DUAN FAN ◽  
QIN-BIN HE ◽  
ZENG-RONG LIU
Keyword(s):  
Dynamical Systems ◽  
Complex Systems ◽  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Stable Equilibrium Point ◽  
Complex Dynamical Systems ◽  
Scientific Communities ◽  
Mathematical Definitions

Recently, the study on the dynamical behavior of complex dynamical systems has become a focal subject in the field of complexity. In particular, the system's adaptability and sensitivity have attracted increasing attention from various scientific communities. In this paper, we focus on some properties of complexity to gain a better understanding of it. Two descriptive mathematical definitions of attractors' adaptability and sensitivity are introduced from the viewpoint of dynamical systems. Then, these new descriptions are applied to analyze the adaptability and sensitivity of stable equilibrium points. In addition, a method is introduced for improving both the adaptability and sensitivity of a system with a stable equilibrium point.

Effects of Low and High Neuron Activation Gradients on the Dynamics of a Simple 3D Hopfield Neural Network

2020 ◽  
Vol 30 (11) ◽  
pp. 2050159 ◽  
Author(s):  
Sami Doubla Isaac ◽  
Z. Tabekoueng Njitacke ◽  
J. Kengne
Keyword(s):  
Neural Network ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Power Spectra ◽  
Hopfield Neural Network ◽  
Basins Of Attraction ◽  
Dynamical Analysis ◽  
Phase Portraits ◽  
Neuron Activation ◽  
Wide Range

In this paper, the effects of low and fast response speeds of neuron activation gradient of a simple 3D Hopfield neural network are explored. It consists of analyzing the effects of low and high neuron activation gradients on the dynamics. By considering an imbalance of the neuron activation gradients, different electrical activities are induced in the network, which enable the occurrence of several nonlinear behaviors. The significant sensitivity of nontrivial equilibrium points to the activation gradients of the first and second neurons relative to that of the third neuron is reported. The dynamical analysis of the model is done in a wide range of the activation gradient of the second neuron. In this range, the model presents areas of periodic behavior, chaotic behavior and periodic window behavior through complex bifurcations. Interesting behaviors such as the coexistences of two, four, six and eight disconnected attractors, as well as the phenomenon of coexisting antimonotonicity, are reported. These singular results are obtained by using nonlinear dynamics analysis tools such as bifurcation diagrams and largest Lyapunov exponents, phase portraits, power spectra and basins of attraction. Finally, some analog results obtained from PSpice-based simulations further verify the numerical results.

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