scholarly journals Influence of Bifurcation Structures Revealed by Refinement of a Nonlinear Conductance in JosephsonJunction Element

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Yuu Miino ◽  
Tetsushi Ueta
Keyword(s):  
Piecewise Linear ◽  
Parameter Plane ◽  
Two Dimensional ◽  
Piecewise Linearization ◽  
Dimensional Parameter ◽  
Current Voltage ◽  
The Stability ◽  
The One ◽  
Bifurcation Structures ◽  
Superconducting Quantum Interferometer

We conduct a bifurcation analysis of a single-junction superconducting quantum interferometer with an external flux. We approximate the current-voltage characteristics of the conductance in the equivalent circuit of the JJ by using two types of functions: a linear function and a piecewise linear (PWL) function. We describe a method to compute the local stability of the solution orbit and to solve the bifurcation problem. As a result, we reveal the bifurcation structure of the systems in a two-dimensional parameter plane. By making a comparison between the linear and PWL cases, we find a difference in the shapes of their bifurcation sets in the two-dimensional parameter plane even though there are no differences in the one-dimensional bifurcation diagrams or the trajectories. As for the influence of piecewise linearization, we discovered that grazing bifurcations terminate the calculation of the local bifurcations, because they drastically change the stability of the periodic orbit.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

The bandcount increment scenario. I. Basic structures

2008 ◽  
Vol 464 (2095) ◽  
pp. 1867-1883 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Basic Structure ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Bifurcation Structures ◽  
Multi Band

Bifurcation structures in two-dimensional parameter spaces formed only by chaotic attractors are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In the first part, we report a novel bifurcation scenario formed by crises bifurcations, which includes multi-band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain.

scholarly journals Stability of Localized Patterns in Neural Fields

2009 ◽  
Vol 21 (4) ◽  
pp. 1125-1144 ◽  
Author(s):  
Konstantin Doubrovinski ◽  
J. Michael Herrmann
Keyword(s):  
Original Problem ◽  
Slight Modification ◽  
Neural System ◽  
Two Dimensional ◽  
Neural Fields ◽  
Localized Solutions ◽  
Localized Patterns ◽  
The Stability ◽  
The One ◽  
Rotationally Invariant

We investigate two-dimensional neural fields as a model of the dynamics of macroscopic activations in a cortex-like neural system. While the one-dimensional case was treated comprehensively by Amari 30 years ago, two-dimensional neural fields are much less understood. We derive conditions for the stability for the main classes of localized solutions of the neural field equation and study their behavior beyond parameter-controlled destabilization. We show that a slight modification of the original model yields an equation whose stationary states are guaranteed to satisfy the original problem and numerically demonstrate that it admits localized noncircular solutions. Typically, however, only periodic spatial tessellations emerge on destabilization of rotationally invariant solutions.

The bandcount increment scenario. II. Interior structures

2008 ◽  
Vol 464 (2097) ◽  
pp. 2247-2263 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Parameter Space ◽  
Complex Interaction ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Self Similar ◽  
Bifurcation Structures

Bifurcation structures in the two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In this second part, we investigate fine substructures nested into the basic structures reported and explained in part I. It is demonstrated that the overall structure of the chaotic domain is caused by a complex interaction of bandcount increment, bandcount adding and bandcount doubling structures, whereby some of them are nested into each other ad infinitum leading to self-similar structures in the parameter space.

GENERALIZATIONS OF THE CHUA EQUATIONS

1992 ◽  
Vol 02 (04) ◽  
pp. 889-909 ◽  
Author(s):  
RAY BROWN
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Type Ii ◽  
Two Dimensional ◽  
Chua’S Circuit ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Chua's Circuit ◽  
The One ◽  
Block Approach

In this paper we present two generalizations of the equations governing Chua’s circuit. In order to obtain the first generalization we simplify Chua’s equations by replacing the piecewise-linear term with a signum function. The resulting simplified system produces a double scroll similar to the one observed experimentally in Chua’s circuit. What is significant about this simplified system is that it can be reduced to what we shall call a two-dimensional single scroll, and from the two-dimensional single scroll we are able to derive a one-dimensional map. This entire derivation is carried out analytically, in contrast to the one-dimensional map analysis that has been carried out for the Lorenz equations which is based on axioms. After we have carried out our analysis for this simplified version of Chua’s equations, we use these equations as a guide to the construction of the first generalization to be presented in this paper. We call this a type-I generalization of Chua’s equations. The generalization consists in using a two-dimensional autonomous flow as a component in a three-dimensional autonomous flow in such a way that the resulting equations will have double scroll attractors similar to those observed experimentally in Chua’s circuit. The value of this generalization is that: (1) it provides a building block approach to the construction of chaotic circuits from simpler two-dimensional components which are not chaotic by themselves. In so doing it provides an insight into how chaotic systems can be built up from simple nonchaotic parts; (2) it illustrates a precise relationship between three-dimensional flows and one-dimensional maps. Of particular significance in this regard is a recent paper of Misiurewicz [1993], which analytically connects the two-dimensional single scroll to the class of unimodal maps, thus providing a framework within which a theory linking unimodal maps to three-dimensional flows may be possible. The second generalization is suggested by considering three-dimensional flows whose only nonlinearities are sigmoid, sgn, or piecewise-linear functions. Clearly, such flows are a generalization of the Chua equations. We call these equations type-II generalization Chua equations. The significance of this direction of investigation is that attractors similar to the Lorenz and Rössler attractors can be produced from type-II generalized Chua equations in a building block approach using only piecewise-linear vector fields. As a result we have a method of producing the Lorenz and Rössler dynamics in a circuit without the use of multipliers. This suggests that the type-II generalized Chua equations are in some sense fundamental in that the dynamics of the three most important autonomous three-dimensional differential equations producing chaos are seen as variations of a single class of equations whose nonlinearities are generalizations of the Chua diode.

SHRIMP STRUCTURE AND ASSOCIATED DYNAMICS IN PARAMETRICALLY EXCITED OSCILLATORS

2006 ◽  
Vol 16 (12) ◽  
pp. 3567-3579 ◽  
Author(s):  
Y. ZOU ◽  
M. THIEL ◽  
M. C. ROMANO ◽  
J. KURTHS ◽  
Q. BI
Keyword(s):  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Two Dimensional ◽  
Recurrence Plots ◽  
Dimensional Parameter ◽  
Parametrically Excited ◽  
Excited System ◽  
Stable Solutions ◽  
Bifurcation Structures

We investigate the bifurcation structures in a two-dimensional parameter space (PS) of a parametrically excited system with two degrees of freedom both analytically and numerically. By means of the Rényi entropy of second order K2, which is estimated from recurrence plots, we uncover that regions of chaotic behavior are intermingled with many complex periodic windows, such as shrimp structures in the PS. A detailed numerical analysis shows that the stable solutions lose stability either via period doubling, or via intermittency when the parameters leave these shrimps in different directions, indicating different bifurcation properties of the boundaries. The shrimps of different sizes offer promising ways to control the dynamics of such a complex system.

A Geometrical Approach Assessing Instability Trends for Galloping

1991 ◽  
Vol 58 (3) ◽  
pp. 784-791 ◽  
Author(s):  
P. Yu ◽  
N. Popplewell ◽  
A. H. Shah
Keyword(s):  
Wind Speed ◽  
Parameter Space ◽  
Degrees Of Freedom ◽  
Critical Conditions ◽  
Geometrical Approach ◽  
Two Dimensional ◽  
Electrical Conductor ◽  
Geometrical Method ◽  
Dimensional Parameter ◽  
The Stability

Although the galloping of an iced electrical conductor has been considered by many researchers, no special attention has been given to the galloping’s sensitivity to alternations in the system’s parameters. A geometrical method is presented in this paper to describe these instability trends and to provide compromises for controlling an instability. The conventional but uncontrollable parameter of the wind speed is chosen as the basis for obtaining the critical conditions under which bifurcations occur for a representative two degrees-of-freedom model. Variations in these critical conditions are found in a two-dimensional parameter space in order to determine the trends for the initiation of galloping as well as to evaluate the stability of the ensuring periodic vibrations.

The bandcount increment scenario. III. Deformed structures

2008 ◽  
Vol 465 (2101) ◽  
pp. 41-57 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Practical Applications ◽  
Chaotic Region ◽  
Self Similar ◽  
Bifurcation Structures

Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely. In a series of three papers, we investigated the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In Part I, the basic structures in the chaotic region are explained by the bandcount increment scenario. In Part II, fine self-similar substructures nested into the bandcount increment scenario are explained by the bandcount-adding and -doubling scenarios, nested into each other ad infinitum. Hereby, we fixed in both previous parts one of the parameters to a non-generic value, and studied the remaining two-dimensional parameter subspace. In this Part III, finally we investigated the structures under variation of this third parameter. Remarkably, this step is the most important with respect to practical applications, since it cannot be expected that these operate exactly at the previously investigated specific value.

scholarly journals A Finite Point Method for Solving the Time Fractional Richards’ Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Xinqiang Qin ◽  
Xin Yang
Keyword(s):  
Point Method ◽  
Richards Equation ◽  
Two Dimensional ◽  
Finite Point ◽  
Basic Principles ◽  
Finite Point Method ◽  
Spatial Derivatives ◽  
Time Fractional Derivative ◽  
The Stability ◽  
The One

In this paper, we propose a numerical method for solving the time fractional Richards’ equation. We first approximate the time fractional derivative of the mentioned equations by a scheme of order O(τ2−α), 0 < a<1; then, we use the finite point method to approximate the spatial derivatives. Before the discrete spatial derivatives, we introduced the basic principles of the finite point method. We solve the one- and two-dimensional versions of these equations using the proposed method. Moreover, the stability properties of the discretized scheme related to time are theoretically analyzed. Numerical results showed the efficiency of the method presented in this paper.

scholarly journals Constructing of Digital Watermark Based on Generalized Fourier Transform

Electronics ◽  
2020 ◽  
Vol 9 (7) ◽  
pp. 1108 ◽  
Author(s):  
Ivanna Dronyuk ◽  
Olga Fedevych ◽  
Natalia Kryvinska
Keyword(s):  
Fourier Transform ◽  
High Stability ◽  
Digital Watermark ◽  
Two Dimensional ◽  
Generalized Fourier Transform ◽  
Trigonometric Functions ◽  
Unauthorized Access ◽  
One Dimensional ◽  
The Stability ◽  
The One

We develop in this paper a method for constructing a digital watermark to protect one-dimensional and two-dimensional signals. The creation of a digital watermark is based on the one-dimensional and two-dimensional generalized Fourier and Hartley transformations and the Ateb-functions as a generalization of trigonometric functions. The embedding of the digital watermark is realized in the frequency domain. The simulation of attacks on protected files is carried out to confirm the stability of the proposed method. Experiments proved the high stability of the developed method conformably to the main types of attacks. An additional built-in digital watermark can be used to identify protected files. The proposed method can be used to support the security of a variety of signals—audio, images, electronic files etc.—to protect them from unauthorized access and as well for identification.

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