scholarly journals Three-Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Bocheng Bao ◽  
Aihuang Hu ◽  
Han Bao ◽  
Quan Xu ◽  
Mo Chen ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Electromagnetic Induction ◽  
Neuron Model ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Neuronal System ◽  
System A ◽  
Dynamical Behaviors ◽  
Electrical Activities

Since the electrical activities of neurons are closely related to complex electrophysiological environment in neuronal system, a novel three-dimensional memristive Hindmarsh–Rose (HR) neuron model is presented in this paper to describe complex dynamics of neuronal activities with electromagnetic induction. The proposed memristive HR neuron model has no equilibrium point but can show hidden dynamical behaviors of coexisting asymmetric attractors, which has not been reported in the previous references for the HR neuron model. Mathematical model based numerical simulations for hidden coexisting asymmetric attractors are performed by bifurcation analyses, phase portraits, attraction basins, and dynamical maps, which just demonstrate the occurrence of complex dynamical behaviors of electrical activities in neuron with electromagnetic induction. Additionally, circuit breadboard based experimental results well confirm the numerical simulations.

Generating Four-Wing Hyperchaotic Attractor and Two-Wing, Three-Wing, and Four-Wing Chaotic Attractors in 4D Memristive System

2017 ◽  
Vol 27 (02) ◽  
pp. 1750027 ◽  
Author(s):  
Ling Zhou ◽  
Chunhua Wang ◽  
Lili Zhou
Keyword(s):  
Chaotic System ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Hyperchaotic System ◽  
Multiple Attractors ◽  
System A ◽  
Dynamical Behaviors ◽  
Memristive System ◽  
Line Of Equilibria

By adding only one smooth flux-controlled memristor into a three-dimensional (3D) pseudo four-wing chaotic system, a new real four-wing hyperchaotic system is constructed in this paper. It is interesting to see that this new memristive chaotic system can generate a four-wing hyperchaotic attractor with a line of equilibria. Moreover, it can generate two-, three- and four-wing chaotic attractors with the variation of a single parameter which denotes the strength of the memristor. At the same time, various coexisting multiple attractors (e.g. three-wing attractors, four-wing attractors and attractors with state transition under the same system parameters) are observed in this system, which means that extreme multistability arises. The complex dynamical behaviors of the proposed system are analyzed by Lyapunov exponents (LEs), phase portraits, Poincaré maps, and time series. An electronic circuit is finally designed to implement the hyperchaotic memristive system.

HYPERBOLIC PLYKIN ATTRACTOR CAN EXIST IN NEURON MODELS

2005 ◽  
Vol 15 (11) ◽  
pp. 3567-3578 ◽  
Author(s):  
VLADIMIR BELYKH ◽  
IGOR BELYKH ◽  
ERIK MOSEKILDE
Keyword(s):  
Neuron Model ◽  
Three Dimensional ◽  
Geometrical Approach ◽  
Hyperbolic Attractor ◽  
Neuron Models ◽  
Neuron System ◽  
Physical Systems ◽  
System A ◽  
Bursting Neurons ◽  
Hyperbolic Attractors

Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincaré map giving an accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined.

Multiscroll Chaotic Sea Obtained from a Simple 3D System Without Equilibrium

2016 ◽  
Vol 26 (02) ◽  
pp. 1650031 ◽  
Author(s):  
Sajad Jafari ◽  
Viet-Thanh Pham ◽  
Tomasz Kapitaniak
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Chaotic Systems ◽  
Poincaré Map ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Equilibrium System ◽  
Frequency Spectra ◽  
Chaotic Flows ◽  
New System

Recently, many rare chaotic systems have been found including chaotic systems with no equilibria. However, it is surprising that such a system can exhibit multiscroll chaotic sea. In this paper, a novel no-equilibrium system with multiscroll hidden chaotic sea is introduced. Besides having multiscroll chaotic sea, this system has two more interesting properties. Firstly, it is conservative (which is a rare feature in three-dimensional chaotic flows) but not Hamiltonian. Secondly, it has a coexisting set of nested tori. There is a hidden torus which coexists with the chaotic sea. This new system is investigated through numerical simulations such as phase portraits, Lyapunov exponents, Poincaré map, and frequency spectra. Furthermore, the feasibility of such a system is verified through circuital implementation.

scholarly journals Smooth Nonlinear Fitting Scheme for Analog Multiplierless Implementation of Hindmarsh-Rose Neuron Model

2021 ◽  
Author(s):  
Jianming Cai ◽  
Han Bao ◽  
Quan Xu ◽  
Zhongyun Hua ◽  
Bocheng Bao
Keyword(s):  
Large Scale ◽  
Neuron Model ◽  
Hardware Implementation ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Implementation Cost ◽  
Hyperbolic Tangent ◽  
Nonlinear Fitting ◽  
Chaotic Bursting ◽  
Electrical Activities

Abstract The Hindmarsh-Rose (HR) neuron model is built to describe the neuron electrical activities. Due to the polynomial nonlinearities, multipliers are required to implement the HR neuron model in analog. In order to avoid the multipliers, this brief presents a novel smooth nonlinear fitting scheme. We first construct two nonlinear fitting functions using the composite hyperbolic tangent functions and then implement an analog multiplierless circuit for the two-dimensional (2D) or three- dimensional (3D) HR neuron model. To exhibit the nonlinear fitting effects, numerical simulations and hardware experiments for the fitted HR neuron model are provided successively. The results show that the fitted HR neuron model with analog multiplierless circuit can display different operation patterns of resting, periodic spiking, and periodic/chaotic bursting, entirely behaving like the original HR neuron model. The analog multiplierless circuit has the advantage of low implementation cost and thereby it might be suitable for the hardware implementation of large-scale neural networks.

scholarly journals Dynamical Behaviors of Coupled Memristor-Based Oscillators with Identical and Different Nonlinearities

2018 ◽  
Vol 2018 ◽  
pp. 1-20
Author(s):  
A. Elsonbaty ◽  
A. Abdelkhalek ◽  
A. Elsaid
Keyword(s):  
Mathematical Models ◽  
Coupled Oscillators ◽  
Perturbation Methods ◽  
Normal Forms ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Numerical Techniques ◽  
Ring Configuration ◽  
Dynamical Behaviors

In this work, the interesting dynamics of coupled nonlinear memristor-based oscillators in ring configuration are explored. The mathematical models are derived to describe the possible cases of employing identical or different nonlinearities. Analytical and numerical techniques, involving perturbation methods, normal forms, phase portraits, and Lyapunov exponents are used to investigate various types of dynamical behaviors along with their stability regions in parameters space. The effects of time-delayed coupling on the proposed system are numerically studied. It is demonstrated that the coupled oscillators show rich dynamics including periodic orbits, quasiperiodicity, two-dimensional, and three-dimensional tori.

COMPLEX DYNAMICS IN PENDULUM EQUATION WITH PARAMETRIC AND EXTERNAL EXCITATIONS II

2006 ◽  
Vol 16 (10) ◽  
pp. 3053-3078 ◽  
Author(s):  
ZHUJUN JING ◽  
JIANPING YANG
Keyword(s):  
Complex Dynamics ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
Maximum Lyapunov Exponent ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Pendulum Equation ◽  
Dynamical Behaviors ◽  
Periodic Sets ◽  
Parametric And External Excitations

This paper (II) is a continuation of "Complex dynamics in pendulum equation with parametric and external excitations (I)." By applying second-order averaging method and Melnikov's method, we obtain the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ∊ν, n = 1, 2, 4 and cannot prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for Ω = nω + ∊ν, n = 3, 5–15 by Melnikov's method, where ν is not rational to ω. However, we show the occurrence of chaos in the averaged and original systems under quasi-periodic perturbation for Ω = nω + ∊ν, n = 3, 5 by numerical simulation. The numerical simulations, include the bifurcation diagram of fixed points, bifurcation diagrams in three- and two-dimensional spaces, homoclinic bifurcation surface, maximum Lyapunov exponent, phase portraits, Poincaré map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors, including period-3 orbits in different chaotic regions, interleaving occurrence of chaotic behaviors and quasi-periodic behaviors, a different kind of interior crisis, jumping behavior of quasi-periodic sets, different nice quasi-periodic attractors, nonchaotic attractors and chaotic attractors, coexistence of three quasi-periodic sets, onset of chaos which occurs more than once for a given external frequency or amplitudes, and quasi-periodic route to chaos. We do not find the period-doubling cascade. The dynamical behaviors under quasi-periodic perturbation are different from that of periodic perturbation.

BIFURCATION AND CHAOS IN THE TINKERBELL MAP

2011 ◽  
Vol 21 (11) ◽  
pp. 3137-3156 ◽  
Author(s):  
SHAOLIANG YUAN ◽  
TAO JIANG ◽  
ZHUJUN JING
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Phase Portraits ◽  
Transient Chaos ◽  
Flip Bifurcation ◽  
Maximum Lyapunov Exponent ◽  
Invariant Circle ◽  
Dynamical Behaviors ◽  
Fold Bifurcation ◽  
Period Doubling Bifurcations

In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained.

scholarly journals The Drag Crisis Phenomenon on an Elite Road Cyclist—A Preliminary Numerical Simulations Analysis in the Aero Position at Different Speeds

2020 ◽  
Vol 17 (14) ◽  
pp. 5003
Author(s):  
Pedro Forte ◽  
Jorge E. Morais ◽  
Henrique P. Neiva ◽  
Tiago M. Barbosa ◽  
Daniel A. Marinho
Keyword(s):  
Numerical Simulations ◽  
Drag Coefficient ◽  
Three Dimensional ◽  
Numerical Code ◽  
Viscous Drag ◽  
System A ◽  
Crisis Phenomenon ◽  
Elite Level ◽  
Computer Fluid Dynamics ◽  
And Performance

The drag crisis phenomenon is the drop of drag coefficient (Cd) with increasing Reynolds number (Re) or speed. The aim of this study was to assess the hypothetical drag crisis phenomenon in a sports setting, assessing it in a bicycle–cyclist system. A male elite-level cyclist was recruited for this research and his competition bicycle, helmet, suit, and shoes were used. A three-dimensional (3D) geometry was obtained with a 3D scan with the subject in a static aero position. A domain with 7 m of length, 2.5 m of width and 2.5 m of height was created around the cyclist. The domain was meshed with 42 million elements. Numerical simulations by computer fluid dynamics (CFD) fluent numerical code were conducted at speeds between 1 m/s and 22 m/s, with increments of 1 m/s. The drag coefficient ranged between 0.60 and 0.95 across different speeds and Re. The highest value was observed at 2 m/s (Cd = 0.95) and Re of 3.21 × 105, whereas the lower Cd was noted at 9 m/s (Cd = 0.60) and 9.63 × 105. A drag crisis was noted between 3 m/s and 9 m/s. Pressure Cd ranged from 0.35 to 0.52 and the lowest value was observed at 3 m/s and the highest at 2 m/s. The viscous drag coefficient ranged between 0.15 and 0.43 and presented a trend decreasing from 4 m/s to 22 m/s. Coaches, cyclists, researchers, and support staff must consider that Cd varies with speed and Re, and the bicycle–cyclist dimensions, shape, or form may affect drag and performance estimations. As a conclusion, this preliminary work noted a drag crisis between 3 m/s and 9 m/s in a cyclist in the aero position.

scholarly journals Generate -Scroll Attractor via Composite Switching Controls

2010 ◽  
Vol 2010 ◽  
pp. 1-9
Author(s):  
Baiyu Ou ◽  
Desheng Liu
Keyword(s):  
Numerical Simulations ◽  
Chaotic Attractor ◽  
Complex Dynamics ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Switched System ◽  
Function Series ◽  
New Type ◽  
Three Dimensional Space

We propose a method for designing chaos generators. We introduce a switched system with three-dimensional space functions for generating a new type of chaotic attractor, and then we introduce saturated function series for generating -scroll chaotic attractor. Moreover, we present some examples with numerical simulations that illustrate the efficiency of our method. The statistic behavior is also discussed, which reveals the regularities in the complex dynamics.

Stability and Hopf Bifurcation Analysis in Hindmarsh–Rose Neuron Model with Multiple Time Delays

2016 ◽  
Vol 26 (11) ◽  
pp. 1650187 ◽  
Author(s):  
Dongpo Hu ◽  
Hongjun Cao
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Time Delays ◽  
Single Neuron ◽  
Neuron Model ◽  
Multiple Time ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Multiple Time Delays ◽  
The Stability

In this paper, the dynamical behaviors of a single Hindmarsh–Rose neuron model with multiple time delays are investigated. By linearizing the system at equilibria and analyzing the associated characteristic equation, the conditions for local stability and the existence of local Hopf bifurcation are obtained. To discuss the properties of Hopf bifurcation, we derive explicit formulas to determine the direction of Hopf bifurcation and the stability of bifurcated periodic solutions occurring through Hopf bifurcation. The qualitative analyses have demonstrated that the values of multiple time delays can affect the stability of equilibrium and play an important role in determining the properties of Hopf bifurcation. Some numerical simulations are given for confirming the qualitative results. Numerical simulations on the effect of delays show that the delays have different scales when the two delay values are not equal. The physiological basis is most likely that Hindmarsh–Rose neuron model has two different time scales. Finally, the bifurcation diagrams of inter-spike intervals of the single Hindmarsh–Rose neuron model are presented. These bifurcation diagrams show the existence of complex bifurcation structures and further indicate that the multiple time delays are very important parameters in determining the dynamical behaviors of the single neuron. Therefore, these results in this paper could be helpful for further understanding the role of multiple time delays in the information transmission and processing of a single neuron.

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