Oscillator Made of Only One Memristor and One Battery

2015 ◽  
Vol 25 (03) ◽  
pp. 1530010 ◽  
Author(s):  
Maheshwar PD. Sah ◽  
Zubaer Ibna Mannan ◽  
Hyongsuk Kim ◽  
Leon Chua
Keyword(s):  
Energy Storage ◽  
Hopf Bifurcation ◽  
Intimate Relationship ◽  
Edge Of Chaos ◽  
Traditional Belief ◽  
Bifurcation Phenomenon ◽  
Circuit Elements ◽  
Electronic Oscillator

Contrary to the traditional belief that at least two energy-storage elements (capacitor and/or inductor) and a locally-active nonlinearity are needed to build an electronic oscillator, this paper presents an oscillator made with only two circuit elements, namely, a memristor and a battery. This simplest of all physical oscillators also serves as a textbook example for explaining the intimate relationship between the super-critical Hopf bifurcation phenomenon and the edge of chaos.

Dynamic Analysis of a Bistable Bi-Local Active Memristor and Its Associated Oscillator System

2018 ◽  
Vol 28 (08) ◽  
pp. 1850105 ◽  
Author(s):  
Hui Chang ◽  
Zhen Wang ◽  
Yuxia Li ◽  
Guanrong Chen
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Theory ◽  
Hysteresis Loops ◽  
Oscillator System ◽  
Local Activity ◽  
Edge Of Chaos ◽  
Bifurcation Phenomenon ◽  
Subcritical Hopf Bifurcation ◽  
New Type ◽  
Pinched Hysteresis

This paper proposes a new type of memristor with two distinct stable pinched hysteresis loops and twin symmetrical local activity domains, named as a bistable bi-local active memristor. A detailed and comprehensive analysis of the memristor and its associated oscillator system is carried out to verify its dynamic behaviors based on nonlinear circuit theory and Hopf bifurcation theory. The local-activity domains and the edge-of-chaos domains of the memristor, which are both symmetric with respect to the origin, are confirmed by utilizing the mathematical cogent theory. Finally, the subcritical Hopf bifurcation phenomenon is identified in the subcritical Hopf bifurcation region of the memristor.

On Generalized Hopf Bifurcations

1984 ◽  
Vol 106 (4) ◽  
pp. 327-334 ◽  
Author(s):  
K. Huseyin ◽  
A. S. Atadan
Keyword(s):  
Hopf Bifurcation ◽  
Additional Condition ◽  
System Parameter ◽  
Transversality Condition ◽  
Existence Condition ◽  
The Other ◽  
Hopf Bifurcations ◽  
Bifurcation Phenomenon ◽  
Bifurcation Phenomena ◽  
Lumped Parameter

Two distinct degenerate Hopf bifurcation phenomena associated with autonomous lumped-parameter systems are explored in great detail via the intrinsic harmonic balancing method. It is assumed that the Hopf’s transversality condition is violated and certain other conditions prevail. In one of the cases, the system exhibits a cusp shape bifurcation path which exists for either positive or negative values of the system parameter. On the other hand, the second case is concerned with a tangential bifurcation phenomenon which may not be exhibited unless an additional condition is satisfied. This existence condition is obtained in the course of analysis. The distinctive feature of the paper is that the results concerning the bifurcating paths and limit cycles are given in general, explicit forms which are expected to be very useful in a variety of applications.

Chua Corsage Memristor Oscillator via Hopf Bifurcation

2016 ◽  
Vol 26 (04) ◽  
pp. 1630009 ◽  
Author(s):  
Zubaer Ibna Mannan ◽  
Hyuncheol Choi ◽  
Hyongsuk Kim
Keyword(s):  
Hopf Bifurcation ◽  
Equivalent Circuit ◽  
Series Expansion ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Operating Point ◽  
Small Signal ◽  
Local Activity ◽  
Edge Of Chaos ◽  
In Series

This paper demonstrates that the Chua Corsage Memristor, when connected in series with an inductor and a battery, oscillates about a locally-active operating point located on the memristor’s DC [Formula: see text]–[Formula: see text] curve. On the operating point, a small-signal equivalent circuit is derived via a Taylor series expansion. The small-signal admittance [Formula: see text] is derived from the small-signal equivalent circuit and the value of inductance is determined at a frequency where the real part of the admittance [Formula: see text] of the small-signal equivalent circuit of Chua Corsage Memristor is zero. Oscillation of the circuit is analyzed via an in-depth application of the theory of Local Activity, Edge of Chaos and the Hopf-bifurcation.

NEURONS ARE POISED NEAR THE EDGE OF CHAOS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250098 ◽  
Author(s):  
LEON CHUA ◽  
VALERY SBITNEV ◽  
HYONGSUK KIM
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Nonlinear Differential Equations ◽  
Complex Conjugate ◽  
Excitation Current ◽  
Hopf Bifurcation Point ◽  
Synaptic Excitation ◽  
Edge Of Chaos ◽  
Subcritical Hopf Bifurcation ◽  
Dc Current

This paper shows the action potential (spikes) generated from the Hodgkin–Huxley equations emerges near the edge of chaos consisting of a tiny subset of the locally active regime of the HH equations. The main result proves that the eigenvalues of the 4 × 4 Jacobian matrix associated with the mathematically intractable system of four nonlinear differential equations are identical to the zeros of a scalar complexity function from complexity theory. Moreover, we show the loci of a pair of complex-conjugate zeros migrate continuously as a function of an externally applied DC current excitation emulating the net synaptic excitation current input to the neuron. In particular, the pair of complex-conjugate zeros move from a subcritical Hopf bifurcation point at low excitation current to a super-critical Hopf bifurcation point at high excitation current. The spikes are generated as the excitation current approaches the vicinity of the edge of chaos, which leads to the onset of the subcritical Hopf bifurcation regime. It follows from this in-depth qualitative analysis that local activity is the origin of spikes.

Analysis of the interaction among weed, inorganic fertilizer and herbivore in paddy ecosystem in fallow season

2017 ◽  
Vol 10 (08) ◽  
pp. 1750120 ◽  
Author(s):  
Meihong Xiang ◽  
Zhaohua Wu ◽  
Tiejun Zhou
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Equation Model ◽  
Inorganic Fertilizer ◽  
Stable Region ◽  
Positive Equilibrium ◽  
Bifurcation Phenomenon ◽  
Bifurcation Phenomena ◽  
Paddy Ecosystem ◽  
Positive Equilibrium Point

Paddy growth is influenced by the amount of inorganic fertilizer in paddy ecosystem in fallow season. To discover the interaction among weed, inorganic fertilizer and herbivore in the system, we put forward a differential equation model and investigate its properties. Results show that the system has a weed and herbivore extinct equilibrium and a herbivore extinct equilibrium. The two equilibria are proven to be unstable using the center manifold method. Under certain conditions, the system also has a positive equilibrium point. We give the stable region and the unstable region of the positive equilibrium point, which are determined by some parameters. We find that the system has the Hopf bifurcation phenomenon, and give the critical value of Hopf bifurcation by taking a system parameter as the bifurcation parameter. By comparing the equilibrium states between a paddy ecosystem with herbivore and one without herbivore, we find that the content of inorganic fertilizer can be improved by putting herbivore into a paddy field. An example is given to illustrate the feasibility of the results. Numerical simulation shows that Hopf bifurcation phenomena exist in the system.

Design of a Low-Frequency Oscillator with PTC Memristor and an Inductor

2016 ◽  
Vol 26 (08) ◽  
pp. 1630021 ◽  
Author(s):  
Vetriveeran Rajamani ◽  
Changju Yang ◽  
Hyongsuk Kim ◽  
Leon Chua
Keyword(s):  
Hopf Bifurcation ◽  
Active Region ◽  
Equivalent Circuit ◽  
Low Frequency ◽  
Negative Resistance ◽  
Small Signal ◽  
Sinusoidal Oscillation ◽  
Frequency Oscillator ◽  
In Series ◽  
Electronic Oscillator

An electronic oscillator circuit is designed by connecting an inductor in series with a locally-active PTC Memristor and a battery. The PTC Memristor is locally active on the negative resistance region of its DC [Formula: see text]–[Formula: see text] curve. A DC operating point [Formula: see text] is chosen on the locally-active region of the PTC Memristor and a small-signal equivalent circuit at [Formula: see text] is derived via Taylor series. The small-signal admittance [Formula: see text] of the composite one-port in Fig. 1 is derived using the small-signal equivalent circuit at [Formula: see text], in series with an inductor whose value is chosen such that [Formula: see text] at some [Formula: see text]. The sinusoidal oscillation computed numerically from this circuit is shown to emerge from a supercritical Hopf bifurcation.

scholarly journals Control of Hopf Bifurcation in Autonomous System Based on Washout Filter

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Wenju Du ◽  
Yandong Chu ◽  
Jiangang Zhang ◽  
Yingxiang Chang ◽  
Jianning Yu ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Equilibrium Points ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Bifurcation Phenomenon ◽  
Controlled Systems ◽  
Washout Filter ◽  
Form Theory ◽  
The Stability

In order to further understand a Lorenz-like system, we study the stability of the equilibrium points and the existence of Hopf bifurcation by center manifold theorem and normal form theory. More precisely, we designed a washout controller such that the equilibriumE0undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibriumE+. Besides, numerical simulation is given to illustrate the theoretical analysis. Finally, two possible electronic circuits are given to realize the uncontrolled and the controlled systems.

New Double Bifurcation of Nilpotent Focus

2021 ◽  
Vol 31 (04) ◽  
pp. 2150053
Author(s):  
Feng Li ◽  
Hongwei Li ◽  
Yuanyuan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Limit Cycles ◽  
Second Order ◽  
Singular Points ◽  
Weak Focus ◽  
Bifurcation Phenomenon ◽  
Planar Systems ◽  
Nilpotent Singular Point

In this paper, a new bifurcation phenomenon of nilpotent singular point is analyzed. A nilpotent focus or center of the planar systems with 3-multiplicity can be broken into two complex singular points and a second order elementary weak focus. Then, two more limit cycles enclosing the second order elementary weak focus can bifurcate through the multiple Hopf bifurcation.

SIMPLEST CHAOTIC CIRCUIT

2010 ◽  
Vol 20 (05) ◽  
pp. 1567-1580 ◽  
Author(s):  
BHARATHWAJ MUTHUSWAMY ◽  
LEON O. CHUA
Keyword(s):  
Energy Storage ◽  
Chaotic Attractor ◽  
The Other ◽  
Chaotic Circuit ◽  
Circuit Topology ◽  
Local Activity ◽  
In Series ◽  
Circuit Elements

A chaotic attractor has been observed with an autonomous circuit that uses only two energy-storage elements: a linear passive inductor and a linear passive capacitor. The other element is a nonlinear active memristor. Hence, the circuit has only three circuit elements in series. We discuss this circuit topology, show several attractors and illustrate local activity via the memristor's DC vM - iM characteristic.

Hopf Bifurcation and Stability Analysis on the Mill Drive System

2011 ◽  
Vol 121-126 ◽  
pp. 1514-1520 ◽  
Author(s):  
Zhi Ying Gao ◽  
Yong Zang ◽  
Tian Han
Keyword(s):  
Hopf Bifurcation ◽  
Time History ◽  
Drive System ◽  
System Stability ◽  
Critical Parameters ◽  
Lumped Mass ◽  
Mass Model ◽  
Bifurcation Phenomenon ◽  
Torsion Stiffness ◽  
Bifurcation And Stability

According to the lumped mass method, the drive system of rolling mill can be simplified as a three-degree-of-freedom spring-mass model. The nonlinear dynamics equations are established considering nonlinear torsion stiffness of connecting shaft and nonlinear friction force between roller and strip, and Hopf bifurcation and critical parameters are analyzed by applying the Hurwitz algorithm criterion. Furthermore, the system stability is numerically simulated through time-history and phase plots. The results indicate that the system motion can be stable in some range of parameters, and the bifurcation phenomenon occurs and the stability may be lost across the critical points. In addition, at different bifurcation points the phase orbit run along diverse skeleton curves of revolution. These conclusions are significant to reveal the mechanism of system bifurcation and stability, and help for optimization of technology condition in practical application.

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