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.

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.

Period doubling boundaries of a bouncing ball

1986 ◽  
Vol 47 (9) ◽  
pp. 1477-1482 ◽  
Author(s):  
N.B. Tufillaro ◽  
T.M. Mello ◽  
Y.M. Choi ◽  
A.M. Albano
Keyword(s):  
Period Doubling ◽  
Bouncing Ball

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.

WEAK DISSIPATION IN A HAMILTONIAN MAP

1994 ◽  
Vol 04 (04) ◽  
pp. 921-932 ◽  
Author(s):  
RAÚL J. MONDRAGÓN C. ◽  
PETER H. RICHTER
Keyword(s):  
Phase Space ◽  
Hamiltonian System ◽  
Periodic Orbits ◽  
Adiabatic Invariance ◽  
Bifurcation Tree ◽  
Bouncing Ball ◽  
Different Types ◽  
Weak Dissipation ◽  
Hamiltonian Map

The dynamics of a bouncing ball reflected off a harmonic spring is investigated, with weak dissipation of three different types. The phase space is found to be organized into a system of tubes that wind around the branches of the bifurcation tree of periodic orbits of the Hamiltonian system. Instead of attraction towards special periodic orbits we observe a kind of piecewise adiabatic invariance of the tubes, with jumps occurring when the branches penetrate each other.

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.

NONLINEAR DYNAMICS OF A BOUNCING BALL DRIVEN BY ELECTRIC FORCES

2003 ◽  
Vol 13 (10) ◽  
pp. 2959-2975 ◽  
Author(s):  
A. KHAYARI ◽  
A. T. PÉREZ
Keyword(s):  
Time Series ◽  
Fractal Structure ◽  
Period Doubling ◽  
Control Parameters ◽  
Conducting Liquid ◽  
Lower Electrode ◽  
Bouncing Ball ◽  
Onset Of Chaos ◽  
Experimental Time Series ◽  
Route To Chaos

This paper is devoted to a theoretical and experimental study of the dynamics of a bouncing ball driven by an electric force. The experimental model consists of a metallic ball immersed in a poorly conducting liquid between two horizontal electrodes. The ball bounces upon the lower electrode as a high voltage is applied between the two plates. The measurement of the time between successive impacts produces a time series, which depends on two control parameters, the amplitude and the frequency of the applied voltage. A theoretical model is proposed, which provides a discrete nonlinear map, and discussed in comparison with the experimental results. It is shown that the system exhibits a period doubling route to chaos and a non-Feigenbaum universal scaling at the onset of chaos. Chaotic motion is investigated using the usual tools: Lyapunov exponents, correlation dimensions and entropies. Fractal structure of the chaotic attractor is also brought to evidence in experimental time series as well as in numerical simulations.

On the scaling properties of two unidirectionally coupled period-doubling systems in the presence of noise

2001 ◽  
Vol 27 (11) ◽  
pp. 960-963
Author(s):  
Yu. V. Gulyaev ◽  
Yu. V. Kapustina ◽  
A. P. Kuznetsov ◽  
S. P. Kuznetsov
Keyword(s):  
Period Doubling ◽  
Scaling Properties
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