Multidimensional period doubling structures

2016 ◽  
Vol 72 (3) ◽  
pp. 391-394
Author(s):  
Jeong-Yup Lee ◽  
Dvir Flom ◽  
Shelomo I. Ben-Abraham
Keyword(s):  
Arbitrary Dimension ◽  
Period Doubling ◽  
Pure Point ◽  
Doubling Sequence ◽  
Straightforward Extension

This paper develops the formalism necessary to generalize the period doubling sequence to arbitrary dimension by straightforward extension of the substitution and recursion rules. It is shown that the period doubling structures of arbitrary dimension are pure point diffractive. The symmetries of the structures are pointed out.

Dynamic Analysis and Adaptive Sliding Mode Controller for a Chaotic Fractional Incommensurate Order Financial System

2017 ◽  
Vol 27 (13) ◽  
pp. 1750198 ◽  
Author(s):  
Ahmad Hajipour ◽  
Hamidreza Tavakoli
Keyword(s):  
Control Method ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Largest Lyapunov Exponent ◽  
Sliding Mode Controller ◽  
Adaptive Sliding Mode Controller ◽  
Uncertain Dynamics ◽  
Route To Chaos ◽  
Doubling Sequence

In this study, the dynamic behavior and chaos control of a chaotic fractional incommensurate-order financial system are investigated. Using well-known tools of nonlinear theory, i.e. Lyapunov exponents, phase diagrams and bifurcation diagrams, we observe some interesting phenomena, e.g. antimonotonicity, crisis phenomena and route to chaos through a period doubling sequence. Adopting largest Lyapunov exponent criteria, we find that the system yields chaos at the lowest order of [Formula: see text]. Next, in order to globally stabilize the chaotic fractional incommensurate order financial system with uncertain dynamics, an adaptive fractional sliding mode controller is designed. Numerical simulations are used to demonstrate the effectiveness of the proposed control method.

scholarly journals The Lorenz model for single-mode homogeneously broadened laser: analytical determination of the unpredictable zone

Open Physics ◽  
2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Samia Ayadi ◽  
Olivier Haeberlé
Keyword(s):  
Analytical Solution ◽  
Analytical Procedure ◽  
Dissipative System ◽  
Period Doubling ◽  
Single Mode ◽  
Analytical Determination ◽  
Pumping Rate ◽  
Fifth Order ◽  
Doubling Sequence

AbstractWe have applied harmonic expansion to derive an analytical solution for the Lorenz-Haken equations. This method is used to describe the regular and periodic self-pulsing regime of the single mode homogeneously broadened laser. These periodic solutions emerge when the ratio of the population decay rate ℘ is smaller than 0:11. We have also demonstrated the tendency of the Lorenz-Haken dissipative system to behave periodic for a characteristic pumping rate “2C P”[7], close to the second laser threshold “2C 2th ”(threshold of instability). When the pumping parameter “2C” increases, the laser undergoes a period doubling sequence. This cascade of period doubling leads towards chaos. We study this type of solutions and indicate the zone of the control parameters for which the system undergoes irregular pulsing solutions. We had previously applied this analytical procedure to derive the amplitude of the first, third and fifth order harmonics for the laser-field expansion [7, 17]. In this work, we extend this method in the aim of obtaining the higher harmonics. We show that this iterative method is indeed limited to the fifth order, and that above, the obtained analytical solution diverges from the numerical direct resolution of the equations.

Convection in a rotating cylindrical annulus. Part 2. Transitions to asymmetric and vacillating flow

1987 ◽  
Vol 174 ◽  
pp. 313-326 ◽  
Author(s):  
A. C. Or ◽  
F. H. Busse
Keyword(s):  
Rayleigh Number ◽  
Rossby Waves ◽  
Period Doubling ◽  
Critical Value ◽  
Coriolis Parameter ◽  
Onset Of Convection ◽  
Cylindrical Annulus ◽  
Propagating Waves ◽  
Annular Layer ◽  
Doubling Sequence

The instabilities of convection columns (also called thermal Rossby waves) in a cylindrical annulus rotating about its axis and heated from the outside are investigated as a function of the Prandtl number P and the Coriolis parameter η*. When this latter parameter is sufficiently large, it is found that the primary solution observed at the onset of convection becomes unstable when the Rayleigh number exceeds its critical value by a relatively small amount. Transitions occur to columnar convection which is non-symmetric with respect to the mid-plane of the small-gap annular layer. Further transitions introduce convection flows that vacillate in time or tend to split the row of columns into an inner and an outer row of separately propagating waves. Of special interest is the regime of non-symmetric convection, which exhibits decreasing Nusselt number with increasing Rayleigh number, and the indication of a period doubling sequence associated with vacillating convection.

scholarly journals CLASSIFICATION OF BIFURCATIONS AND ROUTES TO CHAOS IN A VARIANT OF MURALI–LAKSHMANAN–CHUA CIRCUIT

2002 ◽  
Vol 12 (04) ◽  
pp. 783-813 ◽  
Author(s):  
K. THAMILMARAN ◽  
M. LAKSHMANAN
Keyword(s):  
Period Doubling ◽  
Routes To Chaos ◽  
Rich Variety ◽  
Chua Circuit ◽  
Bifurcation Phenomena ◽  
Simulation Results ◽  
The Rich ◽  
Doubling Sequence ◽  
Two Parameter

We present a detailed investigation of the rich variety of bifurcations and chaos associated with a very simple nonlinear parallel nonautonomous LCR circuit with Chua's diode as its only nonlinear element as briefly reported recently [Thamilmaran et al., 2000]. It is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua (MLC) [Murali et al., 1994]. In our study we have constructed two-parameter phase diagrams in the forcing amplitude-frequency plane, both numerically and experimentally. We point out that under the influence of a periodic excitation a rich variety of bifurcation phenomena, including the familiar period-doubling sequence, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, occur. In addition, we have also observed that the periods of many windows satisfy the familiar Farey sequence. Further, reverse bifurcations, antimonotonicity, remerging chaotic band attractors, and so on, also occur in this system. Numerical simulation results using Poincaré section, Lyapunov exponents, bifurcation diagrams and phase trajectories are found to be in agreement with experimental observations. The chaotic dynamics of this circuit is observed experimentally and confirmed both by numerical and analytical studies as well PSPICE simulation results. The results are also compared with the dynamics of the original MLC circuit with reference to the two-parameter space to show the richness of the present circuit.

THE BOUNCING BALL REVISITED

1990 ◽  
Vol 04 (20) ◽  
pp. 1245-1248 ◽  
Author(s):  
ANITA MEHTA ◽  
J.M. LUCK
Keyword(s):  
Phase Space ◽  
Period Doubling ◽  
Winding Number ◽  
Scaling Properties ◽  
Bouncing Ball ◽  
Temporal Phase ◽  
Vibrating Platform ◽  
Doubling Sequence

We consider a ball under the influence of gravity on a vibrating platform where the ball-platform collisions are completely inelastic. The temporal phase space is seen to be divided into transmitting and absorbing regions, which are responsible for the abrupt termination of a period-doubling sequence and the onset of a locking regime, in which an appropriately defined winding number has intriguing scaling properties.

Higher-Order Spectral Analysis to Detect Power-Frequency Mechanisms in a Driven Chua's Circuit

1997 ◽  
Vol 07 (06) ◽  
pp. 1431-1440 ◽  
Author(s):  
Domine M. W. Leenaerts
Keyword(s):  
Period Doubling ◽  
Peak Frequency ◽  
Higher Order ◽  
Nonlinear Interactions ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Power Frequency ◽  
Higher Order Spectra ◽  
Doubling Sequence ◽  
Dominant Frequencies

Higher-order spectra have been used to investigate nonlinear interactions between frequency modes in a driven Chua's circuit. The spectra show that an energy transfer takes place to the dominant frequencies in the circuit, i.e. the input frequency, the primary peak frequency and the harmonics of both frequencies. Other frequencies couplings become less important. Obviously, powers are (nonlinearly) related at different frequencies. When the circuit undergoes a period doubling sequence to chaos, the gain is increasing.

scholarly journals Theory of weakly nonlinear self-sustained detonations

2015 ◽  
Vol 784 ◽  
pp. 163-198 ◽  
Author(s):  
Luiz M. Faria ◽  
Aslan R. Kasimov ◽  
Rodolfo R. Rosales
Keyword(s):  
Stokes Equations ◽  
Period Doubling ◽  
Spatial Dimension ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Dynamical Characteristics ◽  
Reactive Euler Equations ◽  
Doubling Sequence

We propose a theory of weakly nonlinear multidimensional self-sustained detonations based on asymptotic analysis of the reactive compressible Navier–Stokes equations. We show that these equations can be reduced to a model consisting of a forced unsteady small-disturbance transonic equation and a rate equation for the heat release. In one spatial dimension, the model simplifies to a forced Burgers equation. Through analysis, numerical calculations and comparison with the reactive Euler equations, the model is demonstrated to capture such essential dynamical characteristics of detonations as the steady-state structure, the linear stability spectrum, the period-doubling sequence of bifurcations and chaos in one-dimensional detonations and cellular structures in multidimensional detonations.

scholarly journals On a periodically forced, weakly damped pendulum. Part 3: Vertical forcing

1990 ◽  
Vol 32 (1) ◽  
pp. 42-60 ◽  
Author(s):  
Peter J. Bryant ◽  
John W. Miles
Keyword(s):  
Stable Equilibrium ◽  
Period Doubling ◽  
Mathieu Equation ◽  
The Other ◽  
Vertical Acceleration ◽  
Pendulum Motion ◽  
Stability Boundaries ◽  
Period 2 ◽  
Regions Of Stability ◽  
Doubling Sequence

AbstractWe consider the phase-locked solutions of the differential equation governing planar motion of a weakly damped pendulum forced by a prescribed, vertical acceleration εg sin ωt of its pivot, where ω and t are dimensionless, and the unit of time is the reciprocal of the natural frequency. Resonance curves and stability boundaries are presented for downward and inverted oscillations of periods T, 2T, 4T, …, where T (≡ 2π/ω) is the forcing period. Stable, downward oscillations are found to occur in distinct regions of the (ω, ε) plane, reminiscent of the regions of stability of the Mathieu equation (which describes the equivalent undamped, parametrically excited pendulum motion). The regions are dominated by oscillations of frequencies , each region being bounded on one side by a vertical state at rest in stable equilibrium and on the other side by a symmetry-breaking, period-doubling sequence to chaotic motion. Stable, inverted oscillations are found to occur also in distinct regions of the (ω, ε) plane, the principal oscillation in each region being symmetric with period 2T.

BIFURCATIONS IN HEART RHYTHMS

1995 ◽  
Vol 05 (06) ◽  
pp. 1439-1486 ◽  
Author(s):  
TERESA REE CHAY
Keyword(s):  
Current Pulse ◽  
Antiarrhythmic Drugs ◽  
Sinus Node ◽  
Period Doubling ◽  
Rhythmic Activity ◽  
Class Iii ◽  
History Of ◽  
Reentrant Arrhythmia ◽  
The Right ◽  
Doubling Sequence

Heart rhythms exhibit the following interesting phenomena: First, rhythmic cells such as sinus node cells cease their rhythmic activity when a single brief current pulse of the right magnitude is given to the cell at the right time. Second, a premature pulse can initiate a tachycardia (i.e., reentrant arrhythmia) from a quiescent atrial tissue. This tachycardia can be sustained for hours until another brief current pulse is given. Third, ECG recording sometimes shows alternans in electrical activity prior to ventricular tachycardia. Some suggested that this alternans is the first bifurcation in a period doubling sequence. Fourth, a certain class of antiarrhythmic drugs may increase the chance of sudden cardiac death for patients with a history of myocardial infarction. Fifth, when an "adjustable" tricuspid ring is shortened, the conduction velocity of a reentrant wave becomes oscillatory. When the ring size is shortened further, reentry terminates after several oscillatory cycles. Sixth, sustained reentry arises in a tricuspid ring when Class IC drugs (i.e., which exclusively block the sodium channel) are added. On the other hand, Class III (which exclusively blocks the potassium channel) can terminate sustained reentry in the tricuspid ring. In this tutorial review, we explain how one can utilize a bifurcation analysis to explain all these interesting phenomena involved in heart rhythms.

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