Calculation of stable and unstable periodic orbits in a chopper-fed DC drive

2020 ◽  
Vol 8 (1) ◽  
pp. 43-57
Author(s):  
O. O. Kuznyetsov ◽  
Keyword(s):  
Periodic Orbits ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Nonlinear Phenomena ◽  
Unstable Periodic Orbits ◽  
Border Collision Bifurcations ◽  
Stable Periodic ◽  
Main Period ◽  
Dc Drive ◽  
The Stability

It is well known that electric drives demonstrate various nonlinear phenomena. In particular, a chopper-fed analog DC drive system is characterized by the route to chaotic behavior though period-doubling cascade. Besides, the considered system demonstrates coexistence of several stable periodic modes within the stability boundaries of the main period-1 orbit. We discover the evolution of several periodic orbits utilizing the semi-analytical method based on the Filippov theory for the stability analysis of periodic orbits. We analyze, in particular, stable and unstable period-1, 2, 3 and 4 orbits, as well as independent on stability they are significant for the organization of phase space. We demonstrate, in particular, that the unstable periodic orbits undergo border collision bifurcations; those occur according to several scenarios related to the interaction of different orbits of the same period, including persistence border collision, when a periodic orbit is changed by a different orbit of the same period, and birth or disappearance of a couple of orbits of the same period characterized by different topology.

Periodic orbits and chaos around two black holes

1990 ◽  
Vol 431 (1881) ◽  
pp. 183-202 ◽  
Keyword(s):  
Black Holes ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Classical Problem ◽  
Period Doubling ◽  
Satellite Orbits ◽  
Unstable Periodic Orbits ◽  
Newtonian Case ◽  
Stable Periodic ◽  
Escaping Orbits

We calculate orbits of photons and particles in the relativistic problem of two extreme Reissner-Nordström black holes (fixed). In the case of photons there are three types of non-periodic orbits, namely orbits falling into the black holes M 1 and M 2 (types I and II), and orbits escaping to infinity (III). The various types of orbits are separated by orbits asymptotic to the three main types of (unstable) periodic orbits: ( a ) around M 1 , ( b ) around M 2 and ( c ) around both M 1 and M 2 . Between two non-periodic orbits of two different types there are orbits of the third type. The initial conditions of the three types of orbits form three Cantor sets, and this fact is a manifestation of chaos. The role of higher-order periodic orbits is explained. In the case of particles the situation is similar in the parabolic and hyperbolic cases. However, in the elliptic case the orbits are contained inside a curve of zero velocity, and there are no escaping orbits. Instead we have orbits trapped around stable periodic orbits (IV) and stochastic orbits not falling on the black holes M 1 and M 2 (V). The stable periodic orbits become unstable by period doubling, and infinite period doublings lead to chaos. The newtonian limit is an integrable problem (essentially the same as the classical problem of two fixed centres). The periodic orbits of type ( c ) are topologically similar to the corresponding relativistic orbits, but they differ considerably numerically. We prove that in the newtonian case there are no satellite orbits around M 1 or M 2 (of type ( a ) or ( b )). The post-newtonian case is non-integrable. In this case there are in general orbits of all three types ( a ), ( b ) and ( c ).

ZERO-CROSS INSTANTANEOUS STATE SETTING FOR CONTROL OF A BIFURCATING H-BRIDGE INVERTER

2007 ◽  
Vol 17 (10) ◽  
pp. 3571-3575 ◽  
Author(s):  
SATOSHI AKATSU ◽  
HIROYUKI TORIKAI ◽  
TOSHIMICHI SAITO
Keyword(s):  
Pulse Width Modulation ◽  
Control Method ◽  
Period Doubling ◽  
Current Mode ◽  
Nonlinear Phenomena ◽  
Unstable Periodic Orbits ◽  
Bifurcation And Chaos ◽  
Zero Crossing ◽  
Modulation Signal ◽  
A Current

This paper studies stabilization of low-period unstable periodic orbits (UPOs) in a simplified model of a current mode H-bridge inverter. The switching of the inverter is controlled by pulse-width modulation signal depending on the sampled inductor current. The inverter can exhibit rich nonlinear phenomena including period doubling bifurcation and chaos. Our control method is realized by instantaneous opening of inductor at a zero-crossing moment of an objective UPO and can stabilize the UPO instantaneously as far as the UPO crosses zero in principle. Typical system operations can be confirmed by numerical experiments.

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.

Nonlinear Dynamic Analysis of a Simplest Fractional-Order Delayed Memristive Chaotic System

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Wei Hu ◽  
Dawei Ding ◽  
Nian Wang
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Behavior ◽  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Transversality Condition ◽  
Memristive System ◽  
Route To Chaos ◽  
The Stability

A simplest fractional-order delayed memristive chaotic system is investigated in order to analyze the nonlinear dynamics of the system. The stability and bifurcation behaviors of this system are initially investigated, where time delay is selected as the bifurcation parameter. Some explicit conditions for describing the stability interval and the transversality condition of the emergence for Hopf bifurcation are derived. The period doubling route to chaos behaviors of such a system is discussed by using a bifurcation diagram, a phase diagram, a time-domain diagram, and the largest Lyapunov exponents (LLEs) diagram. Specifically, we study the influence of time delay on the chaotic behavior, and find that when time delay increases, the transitions from one cycle to two cycles, two cycles to four cycles, and four cycles to chaos are observed in this system model. Corresponding critical values of time delay are determined, showing the lowest orders for chaos in the fractional-order delayed memristive system. Finally, numerical simulations are provided to verify the correctness of theoretical analysis using the modified Adams–Bashforth–Moulton method.

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.

BIFURCATIONS AND CHAOS IN CELLULAR NEURAL NETWORKS

2003 ◽  
Vol 12 (04) ◽  
pp. 417-433 ◽  
Author(s):  
M. BIEY ◽  
P. CHECCO ◽  
M. GILLI
Keyword(s):  
Neural Networks ◽  
Bifurcation Analysis ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Cellular Neural Networks ◽  
Two Dimensional ◽  
First Order ◽  
Stable Periodic ◽  
Torus Breakdown

The dynamic behavior of first-order autonomous space invariant cellular neural networks (CNNs) is investigated. It is shown that complex dynamics may occur in very simple CNN structures, described by two-dimensional templates that present only vertical and horizontal couplings. The bifurcation processes are analyzed through the computation of the limit cycle Floquet's multipliers, the evaluation of the Lyapunov exponents and of the signal spectra. As a main result a detailed and accurate two-dimensional bifurcation diagram is reported. The diagram allows one to distinguish several regions in the parameter space of a single CNN. They correspond to stable, periodic, quasi-periodic, and chaotic behavior, respectively. In particular it is shown that chaotic regions can be reached through two different routes: period doubling and torus breakdown. We remark that most practical CNN implementations exploit first order cells and space-invariant templates: so far only a few examples of complex dynamics and no complete bifurcation analysis have been presented for such networks.

A Four-Leaf Chaotic Attractor of a Three-Dimensional Dynamical System

2015 ◽  
Vol 25 (02) ◽  
pp. 1530003 ◽  
Author(s):  
Tomoyuki Miyaji ◽  
Hisashi Okamoto ◽  
Alex D. D. Craik
Keyword(s):  
Dynamical System ◽  
Three Dimensional ◽  
Period Doubling ◽  
Numerical Verification ◽  
Unstable Periodic Orbits ◽  
Verification Method ◽  
Stable Periodic ◽  
Approach Infinity ◽  
Dimensional Dynamical System ◽  
Autonomous Dynamical System

A three-dimensional autonomous dynamical system proposed by Pehlivan is untypical in simultaneously possessing both unbounded and chaotic solutions. Here, this topic is studied in some depth, both numerically and analytically. We find, by standard methods, that four-leaf chaotic orbits result from a period-doubling cascade; we identify unstable fixed points and both stable and unstable periodic orbits; and we examine how initial data determines whether orbits approach infinity or a stable periodic orbit. Further, we describe and apply a strict numerical verification method that rigorously proves the existence of sequences of period doublings.

Bifurcations of Periodic Orbits of a One-Dimensional Pre-Compressed Granular Array

2018 ◽  
Author(s):  
Gizem Dilber Acar ◽  
Balakumar Balachandran
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Bifurcation Point ◽  
Granular Media ◽  
Period Doubling ◽  
Initial Step ◽  
Hertzian Contact ◽  
One Dimensional ◽  
The Stability ◽  
External Forces

Bifurcations of periodic orbits of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is considered without any damping or external forces. By using the Hertzian contact model, and confining the system’s total energy to a certain level, changes in in-phase periodic orbit are studied for various pre-compression levels. At a certain pre-compression level, symmetry breaking and period doubling occur, and an asymmetric period-two orbit emerges from the in-phase periodic orbit. Floquet analysis is conducted to study the stability of the in-phase periodic solution, and to detect the bifurcation location. Although the trajectory of period-two orbit is close to the in-phase orbit at the bifurcation point, the asymmetry of the period-two orbit becomes more pronounced as one moves away from the bifurcation point. This work is meant to serve as an initial step towards understanding how pre-compression may introduce qualitative changes in system dynamics of granular media.

CHAOS CONTROL OF A MODIFIED COUPLED DYNAMOS SYSTEM

2007 ◽  
Vol 21 (26) ◽  
pp. 4593-4610 ◽  
Author(s):  
XING-YUAN WANG ◽  
XIANG-JUN WU
Keyword(s):  
Numerical Simulations ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Direct Method ◽  
Equilibrium Points ◽  
Unstable Equilibrium ◽  
Unstable Periodic Orbits ◽  
Lyapunov Direct Method

This paper studies the problem of controlling the chaotic behavior of a modified coupled dynamos system. Two different methods, feedback and non-feedback methods, are used to control chaos in the modified coupled dynamos system. Based on the Lyapunov direct method and Routh–Hurwitz criterion, the conditions suppressing chaos to unstable equilibrium points or unstable periodic orbits (limit cycles) are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the two different methods.

scholarly journals ON THE STABILIZATION OF PERIODIC ORBITS FOR DISCRETE TIME CHAOTIC SYSTEMS BY USING SCALAR FEEDBACK

2007 ◽  
Vol 17 (12) ◽  
pp. 4431-4442 ◽  
Author(s):  
ÖMER MORGÜL
Keyword(s):  
Periodic Orbits ◽  
Discrete Time ◽  
Chaotic Systems ◽  
Delayed Feedback ◽  
Stabilization Problem ◽  
Unstable Periodic Orbits ◽  
Scalar Input ◽  
Simulation Results ◽  
The Stability ◽  
Stability Results

In this paper we consider the stabilization problem of unstable periodic orbits of discrete time chaotic systems by using a scalar input. We use a simple periodic delayed feedback law and present some stability results. These results show that all hyperbolic periodic orbits as well as some nonhyperbolic periodic orbits can be stabilized with the proposed method by using a scalar input, provided that some controllability or observability conditions are satisfied. The stability proofs also lead to the possible feedback gains which achieve stabilization. We will present some simulation results as well.

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