ANALYTICAL PREDICTION OF THE TWO FIRST PERIOD-DOUBLINGS IN A THREE-DIMENSIONAL SYSTEM

2000 ◽  
Vol 10 (06) ◽  
pp. 1497-1508 ◽  
Author(s):  
M. BELHAQ ◽  
M. HOUSSNI ◽  
E. FREIRE ◽  
A. J. RODRÍGUEZ-LUIS
Keyword(s):  
Periodic Orbit ◽  
Critical Parameter ◽  
Order Approximation ◽  
Three Dimensional ◽  
Period Doubling ◽  
Dimensional System ◽  
Analytical Prediction ◽  
Parameter Values ◽  
Three Dimensional System ◽  
Period Doubling Bifurcations

Analytical study of the two first period-doubling bifurcations in a three-dimensional system is reported. The multiple scales method is first applied to construct a higher-order approximation of the periodic orbit following Hopf bifurcation. The stability analysis of this periodic orbit is then performed in terms of Floquet theory to derive the critical parameter values corresponding to the first and second period-doubling bifurcations. By introducing suitable subharmonic components in the first order of the multiple scale analysis the two critical parameter values are obtained simultaneously solving analytically the resulting system of two algebraic equations. Comparisons of analytic predictions to numerical simulations are also provided.

Second period-doubling in a three-dimensional system

1999 ◽  
Vol 26 (2) ◽  
pp. 123-128 ◽  
Author(s):  
M. Belhaq ◽  
E. Freire ◽  
M. Houssni ◽  
A.J. Rodríguez-Luis
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Dimensional System ◽  
Second Period ◽  
Three Dimensional System

The Dynamics of Symmetric Bimodal Maps

1997 ◽  
Vol 07 (12) ◽  
pp. 2735-2744 ◽  
Author(s):  
Thomas Lofaro
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Period Doubling ◽  
General Setting ◽  
The Family ◽  
Pitchfork Bifurcations ◽  
Parameter Values ◽  
Kneading Theory ◽  
Period Doubling Bifurcations

The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, fα are discussed. We show that for a large class of parameter values the dynamics of fα can be described via an identification with a unimodal map uα. In this parameter regime, a periodic orbit of period 2n + 1 of uα corresponds to a periodic orbit of period 4n + 2 for fα. A periodic orbit of period 2n of uα corresponds to a pair of distinct periodic orbits also of period 2n for fα. In a more general setting we describe the genealogy of periodic orbits in the family fα using symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.

ANALYTICS OF HOMOCLINIC BIFURCATIONS IN THREE-DIMENSIONAL SYSTEMS

2002 ◽  
Vol 12 (11) ◽  
pp. 2479-2486 ◽  
Author(s):  
MOHAMED BELHAQ ◽  
FAOUZI LAKRAD
Keyword(s):  
Periodic Orbit ◽  
Analytical Approach ◽  
Critical Parameter ◽  
Multiple Scales ◽  
Three Dimensional ◽  
Specific System ◽  
Homoclinic Bifurcations ◽  
Hyperbolic Fixed Point ◽  
Parameter Values ◽  
Multiple Scales Perturbation Method

An analytical approach to determine critical parameter values of homoclinic bifurcations in three-dimensional systems is reported. The homoclinic orbit is supposed to be a limit of a unique periodic orbit. Hence, the multiple scales perturbation method is performed to construct an approximation of the periodic solution and its frequency. Then, two simple criteria are used. The first criterion is based on the collision between the periodic and the hyperbolic fixed point involved in the bifurcation. The second uses the infinity condition of the period of the periodic orbit. For illustration a specific system is investigated.

Inner-product theory calculations for some rational potentials in a three-dimensional system

1996 ◽  
Vol 74 (1-2) ◽  
pp. 4-9
Author(s):  
M. R. M. Witwit
Keyword(s):  
Energy Levels ◽  
Three Dimensional ◽  
Inner Product ◽  
Dimensional System ◽  
Product Technique ◽  
Special Cases ◽  
Wide Range ◽  
Perturbation Parameters ◽  
Three Dimensional System ◽  
Range Of Values

The energy levels of a three-dimensional system are calculated for the rational potentials,[Formula: see text]using the inner-product technique over a wide range of values of the perturbation parameters (λ, g) and for various eigenstates. The numerical results for some special cases agree with those of previous workers where available.

The influence of mean shear on Alfvén-internal wave propagation

1976 ◽  
Vol 15 (2) ◽  
pp. 197-222
Author(s):  
R. J. Hartman
Keyword(s):  
Three Dimensional ◽  
Dimensional System ◽  
Linear Wave ◽  
Mean Flow ◽  
Wave Processes ◽  
Initial Value ◽  
Initial Wave ◽  
Wave Phenomena ◽  
Three Dimensional System

This paper uses the general solution of the linearized initial-value problem for an unbounded, exponentially-stratified, perfectly-conducting Couette flow in the presence of a uniform magnetic field to study the development of localized wave-type perturbations to the basic flow. The two-dimensional problem is shown to be stable for all hydrodynamic Richardson numbers JH, positive and negative, and wave packets in this flow are shown to approach, asymptotically, a level in the fluid (the ‘isolation level’) which is a smooth, continuous, function of JH that is well defined for JH < 0 as well as JH > 0. This system exhibits a rich complement of wave phenomena and a variety of mechanisms for the transport of mean flow kinetic and potential energy, via linear wave processes, between widely-separated regions of fluid; this in addition to the usual mechanisms for the absorption of the initial wave energy itself. The appropriate three-dimensional system is discussed, and the role of nonlinearities on the development of localized disturbances is considered.

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