ON THE GLOBAL BEHAVIOR OF SELF-OSCILLATORY SYSTEMS

1993 ◽  
Vol 03 (01) ◽  
pp. 195-200 ◽  
Author(s):  
A.D. MOROZOV
Keyword(s):  
Hamiltonian System ◽  
Invariant Torus ◽  
Degrees Of Freedom ◽  
Systems Approach ◽  
Global Behavior ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
System Parameters ◽  
Oscillatory Systems ◽  
Good Agreement

The problem of the global behavior of the system with 3/2 degrees of freedom is considered in this paper. The results of the theoretical investigation of the case when the systems approach a nonlinear Hamiltonian system are presented. Using, for example, the Duffing-van der Pol equation, computer analysis gives good agreement with theory when the external force has moderate amplitude values (“miracle of a small parameter”). The bifurcations of the invariant torus and resonances are analyzed as a function of the system parameters.

A theory for phase locking of respiration in cats to a mechanical ventilator

1984 ◽  
Vol 246 (3) ◽  
pp. R311-R320 ◽  
Author(s):  
G. A. Petrillo ◽  
L. Glass
Keyword(s):  
Nonlinear Oscillator ◽  
Phase Locking ◽  
Experimental Studies ◽  
Mechanical Ventilator ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Mechanically Ventilated ◽  
Wide Range ◽  
Forced Nonlinear Oscillator ◽  
Good Agreement

A mathematical model describing the Hering-Breuer reflexes in mechanically ventilated cats is developed. There is good agreement between the properties of the model and experimental studies performed over a wide range of frequencies and volumes of the mechanical ventilator. There is a correspondence between the model and a periodically forced nonlinear oscillator, similar to the van der Pol equation. Brain stem mechanisms underlying the entrainment are discussed.

scholarly journals Perturbation solutions of fifth order oscillatory nonlinear systems

2011 ◽  
Vol 16 (2) ◽  
pp. 123-134
Author(s):  
M. Ali Akbar ◽  
Sk. Tanzer Ahmed Siddique
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Weakly Nonlinear ◽  
Physical Systems ◽  
Oscillatory Systems ◽  
Engineering Problems ◽  
Kbm Method ◽  
Fifth Order ◽  
Method Show ◽  
Good Agreement

Oscillatory systems play an important role in the nature. Many engineering problems and physical systems of fifth degrees of freedom are oscillatory and their governing equations are fifth order nonlinear differential equations. To investigate the solution of fifth order weakly nonlinear oscillatory systems, in this article the Krylov–Bogoliubov–Mitropolskii (KBM) method has been extended and desired solution is found. An example is solved to illustrate the method. The results obtain by the extended KBM method show good agreement with those obtained by numerical method.

scholarly journals The frequency of Kozai–Lidov disc oscillation driven giant outbursts in Be/X-ray binaries

2019 ◽  
Vol 489 (2) ◽  
pp. 1797-1804 ◽  
Author(s):  
Rebecca G Martin ◽  
Alessia Franchini
Keyword(s):  
Orbital Period ◽  
Time Scale ◽  
Orbital Plane ◽  
Material Flows ◽  
X Ray ◽  
System Parameters ◽  
Be Star ◽  
Chaotic Nature ◽  
X Ray Binaries ◽  
Good Agreement

ABSTRACT Giant outbursts of Be/X-ray binaries may occur when a Be-star disc undergoes strong eccentricity growth due to the Kozai–Lidov (KL) mechanism. The KL effect acts on a disc that is highly inclined to the binary orbital plane provided that the disc aspect ratio is sufficiently small. The eccentric disc overflows its Roche lobe and material flows from the Be star disc over to the companion neutron star causing X-ray activity. With N-body simulations and steady state decretion disc models we explore system parameters for which a disc in the Be/X-ray binary 4U 0115+634 is KL unstable and the resulting time-scale for the oscillations. We find good agreement between predictions of the model and the observed giant outburst time-scale provided that the disc is not completely destroyed by the outburst. This allows the outer disc to be replenished between outbursts and a sufficiently short KL oscillation time-scale. An initially eccentric disc has a shorter KL oscillation time-scale compared to an initially circular orbit disc. We suggest that the chaotic nature of the outbursts is caused by the sensitivity of the mechanism to the distribution of material within the disc. The outbursts continue provided that the Be star supplies material that is sufficiently misaligned to the binary orbital plane. We generalize our results to Be/X-ray binaries with varying orbital period and find that if the Be star disc is flared, it is more likely to be unstable to KL oscillations in a smaller orbital period binary, in agreement with observations.

Estimation of the Thermodynamic Properties of Branched Hydrocarbons

2000 ◽  
Vol 122 (3) ◽  
pp. 147-152 ◽  
Author(s):  
Hui He ◽  
Mohamad Metghalchi ◽  
James C. Keck
Keyword(s):  
Thermodynamic Properties ◽  
Degrees Of Freedom ◽  
Statistical Thermodynamic ◽  
Motion Modes ◽  
Branched Alkanes ◽  
Branched Hydrocarbons ◽  
Wide Range ◽  
On Line ◽  
Kinetic Calculations ◽  
Good Agreement

A simple model has been developed to estimate the sensible thermodynamic properties such as Gibbs free energy, enthalpy, heat capacity, and entropy of hydrocarbons over a wide range of temperatures with special attention to the branched molecules. The model is based on statistical thermodynamic expressions incorporating translational, rotational and vibrational motions of the atoms. A method to determine the number of degrees of freedom for different motion modes (bending and torsion) has been established. Branched rotational groups, such as CH3 and OH, have been considered. A modification of the characteristic temperatures for different motion mode has been made which improves the agreement with the exact values for simple cases. The properties of branched alkanes up to 2,3,4,-trimthylpentane have been calculated and the results are in good agreement with the experimental data. A relatively small number of parameters are needed in this model to estimate the sensible thermodynamic properties of a wide range of species. The model may also be used to estimate the properties of molecules and their isomers, which have not been measured, and is simple enough to be easily programmed as a subroutine for on-line kinetic calculations. [S0195-0738(00)00902-X]

scholarly journals On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

2014 ◽  
Vol 24 (05) ◽  
pp. 1450061 ◽  
Author(s):  
Albert D. Morozov ◽  
Olga S. Kostromina
Keyword(s):  
Saddle Point ◽  
Limit Cycles ◽  
Small Neighborhood ◽  
Integrable Equation ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Periodic Perturbations ◽  
Autonomous Equation ◽  
Nonautonomous Equation ◽  
Time Periodic

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2823-2869 ◽  
Author(s):  
Z. E. MUSIELAK ◽  
D. E. MUSIELAK
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
General Rules ◽  
Van Der Pol Oscillators ◽  
Control And Synchronization

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

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