HOPF BIFURCATION FROM LINES OF EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

2010 ◽  
Vol 20 (02) ◽  
pp. 437-450 ◽  
Author(s):  
MARCELO MESSIAS ◽  
CRISTIANE NESPOLI ◽  
VANESSA A. BOTTA
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Fixed Set ◽  
Electronic Element ◽  
Stable Periodic ◽  
Parameter Values

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions.

scholarly journals THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION

2011 ◽  
Vol 21 (08) ◽  
pp. 2321-2330 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
G. CONTOPOULOS
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Transition Point ◽  
Initial Conditions ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Surface Of Section ◽  
Autonomous Hamiltonian System ◽  
Hamiltonian Hopf Bifurcation ◽  
Stable Periodic

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

On Discontinuous Piecewise Linear Models for Memristor Oscillators

2017 ◽  
Vol 27 (06) ◽  
pp. 1730022 ◽  
Author(s):  
Andrés Amador ◽  
Emilio Freire ◽  
Enrique Ponce ◽  
Javier Ros
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Linear Models ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Infinite Family ◽  
Closed Surfaces ◽  
The Rich ◽  
Piecewise Linear Models

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.

scholarly journals Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

2017 ◽  
Vol 27 (12) ◽  
pp. 1730042 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Simple Type ◽  
Asymptotically Stable ◽  
Sliding Bifurcation ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Smooth System ◽  
Structurally Stable

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

STABILIZATION OF CHAOTIC DYNAMICS OF ONE-DIMENSIONAL MAPS BY A CYCLIC PARAMETRIC TRANSFORMATION

1996 ◽  
Vol 06 (04) ◽  
pp. 725-735 ◽  
Author(s):  
ALEXANDER Yu. LOSKUTOV ◽  
VALERY M. TERESHKO ◽  
KONSTANTIN A. VASILIEV
Keyword(s):  
Periodic Orbits ◽  
Chaotic Dynamics ◽  
Chaotic Maps ◽  
Logistic Map ◽  
Exponential Map ◽  
One Dimensional ◽  
Stable Periodic ◽  
One Dimensional Maps ◽  
Parameter Values ◽  
Parametric Transformation

We consider one-dimensional maps, the logistic map and an exponential map, in those sets of parameter values which correspond to their chaotic dynamics. It is proven that such dynamics may be stabilized by a certain cyclic parametric transformation operating strictly within the chaotic set. The stabilization is a result of the creation of stable periodic orbits in the initially chaotic maps. The period of these stable orbits is a multiple of the period of the cyclic transformation. It is shown that stabilized behavior cannot be destroyed by a weak noise smearing of the required parameter values. The regions where the behavior stabilization takes place are numerically estimated. Periods of the created stabile periodic orbits are calculated.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

scholarly journals Periodic orbits relevant for the formation of inner rings in barred galaxies

1985 ◽  
Vol 106 ◽  
pp. 543-544
Author(s):  
M. Michalodimitrakis ◽  
Ch. Terzides
Keyword(s):  
Periodic Orbits ◽  
Three Dimensional ◽  
Particle Trapping ◽  
Barred Galaxies ◽  
Future Paper ◽  
Stability Of Periodic Orbits ◽  
Wide Range ◽  
The Galaxy ◽  
Stable Periodic ◽  
The Stability

The study of orbits of a test particle in the gravitational field of a model barred galaxy is a first step toward the understanding of the origin of the morphological characterstics observed in real barred galaxies. In this paper we confine our attention to the inner rings. Inner rings are a very common characteristic of barred galaxies. They are narrow, round or slightly elongated along the bar (with typical axial ratios from 0.7 to near 1.0), and of the same size as the bar. A first step to test the old hypothesis that inner rings consist of stars trapped near stable periodic orbits would be a study of particle trapping around periodic orbits encircling the bar. Such a study is contained in the work of several authors (Danby 1965, de Vaucouleurs and Freeman 1972, Michalodimitrakis 1975, Contopoulos and Papayannopoulos 1980, Athanassoula et al. 1983). In the above works the stability of periodic orbits was studied with respect to perturbations which lie on the plane of motion z = 0 (planar stability). To ensure the possibility of formation of rings, a study of stability with respect to perturbations perpendicular to the plane of motion (vertical stability) is necessary. In this paper we investigate the properties of periodic orbits which we believe to be relevant for the inner-ring problem using a sufficiently general model for the galaxy and sets of values for the parameters which cover a wide range of different possible cases. We also study the stability, planar and vertical, with respect to large perturbations in order to estimate the extent of particle trapping. A detailed numerical investigation of three-dimensional periodic orbits will be given in a future paper.

scholarly journals Orbital Structure of the Two Fixed Centres Problem

1999 ◽  
Vol 172 ◽  
pp. 463-464
Author(s):  
A. Cordero ◽  
J. Martínez Alfaro ◽  
P. Vindel
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Artificial Satellite ◽  
Equilibrium Points ◽  
Integrable Hamiltonian System ◽  
Three Dimensional ◽  
First Integral ◽  
Material Point ◽  
Dimensional Sphere

The set of orbits of the Two Fixed Centres problem has been known for a long time (Chartier, 1902, 1907; Pars, 1965), since it is an integrable Hamiltonian system.We consider a plane that contains the fixed masses. Denote by φ the angle denned by this plane and the one that contains also the third body. The momentum pφ is a first integral of the system and when pφ is different from zero, the manifold generated by the generalized coordinates and momenta are two copies of the three-dimensional sphere S3. If pφ = 0, that is to say when the planet crosses the line joining both suns, the motion is restricted to a planar one. All the equilibrium points appears in this case and therefore the phase spaces are more complex. We restrict our attention to this case which has two degrees of freedom.It is again a Bott-integrable Hamiltonian system. The set of periodic orbits of this systems can be studied from a subset of them, the Non-Singular Morse-Smale type orbits (see Casasayas, 1992). It is proved in Campos (1997) that a small perturbation of a Bott-integrable Hamiltonian system transforms it into a Non-Singular Morse-Smale system. The NMS periodic orbits belong to both the NMS system and the Hamiltonian one. Moreover, The NMS p.o. can be continued to nearly Hamiltonian systems. For instance, in our case to the Restricted Three Body Problem and in the study of the motion of a material point moving inside the gravitational field generated by two stars. This approximation is also useful when the motion of an artificial satellite around a spheroidal body is considered.

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