scholarly journals Two Nested Limit Cycles in Two-Species Reactions

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1658
Author(s):  
Ilona Nagy ◽  
Valery G. Romanovski ◽  
János Tóth
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Reaction Rate ◽  
Rate Coefficients ◽  
Reaction Step ◽  
Third Order ◽  
Outer Limit ◽  
Symbolic Calculations ◽  
Focus Type ◽  
Fast Oscillations

We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential system. Symbolic calculations were carried out using the Mathematica computer algebra system, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each have two limit cycles surrounding the stationary point of focus type in the positive quadrant. In the case of Model 1, the outer limit cycle is stable and the inner one is unstable, which appears in a supercritical Hopf bifurcation. Moreover, the oscillations in a neighborhood of the outer limit cycle are slow-fast oscillations. In the case of Model 2, the outer limit cycle is unstable and the inner one is stable. With another set of parameters, the outer limit cycle can be made stable and the inner one unstable.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

Methyl-Perfluoroheptene-Ethers (CH3OC7F13): Measured OH Radical Reaction Rate Coefficients for Several Isomers and Enantiomers and Their Atmospheric Lifetimes and Global Warming Potentials

2014 ◽  
Vol 48 (9) ◽  
pp. 4954-4962 ◽  
Author(s):  
Aaron M. Jubb ◽  
Tomasz Gierczak ◽  
Munkhbayar Baasandorj ◽  
Robert L. Waterland ◽  
James B. Burkholder
Keyword(s):  
Global Warming ◽  
Reaction Rate ◽  
Radical Reaction ◽  
Rate Coefficients ◽  
Oh Radical ◽  
Global Warming Potentials

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

scholarly journals The alcoholysis of 1,2,2-trimethylpropyl-methylfluorophosphonate

2000 ◽  
Vol 65 (12) ◽  
pp. 857-866
Author(s):  
Mladjen Micevic ◽  
Slobodan Petrovic
Keyword(s):  
Kinetic Data ◽  
Reaction Rate ◽  
Frequency Factor ◽  
Reaction Rate Constant ◽  
Order Reaction ◽  
Activation Entropy ◽  
Second Order Reaction ◽  
Third Order ◽  
First Order ◽  
Kinetics Of

The alcoholysis of 1,2,2-trimethylpropyl-methylfluorophosphonate (soman) was examined with a series of alkoxides and in corresponding alcohols: methanol, ethanol, 1-propanol, 2-propanol, 2-methoxyethanol and 2-ethoxyethanol. Soman reacts with the used alkoxides in a second order reaction, first order in each reactant. The kinetics of the reaction between 1,2,2-trimethylpropyl-methylfluorophosphonate and ethanol in the presence of diethylenetriamine was also examined. A third order reaction rate constant was calculated, first order in each reactant. The activation energy, frequency factor and activation entropy were determined on the basis of the kinetic data.

Determination of the Limit Cycle by He’s Parameter-Expansion for Oscillators in a u3 / (1 + u2) Potential

2007 ◽  
Vol 62 (7-8) ◽  
pp. 396-398 ◽  
Author(s):  
Li-Na Zhang ◽  
Lan Xu
Keyword(s):  
Exact Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Expansion Method ◽  
Nonlinear Oscillators ◽  
Mathematical Tool ◽  
Parameter Expansion ◽  
Parameter Expansion Method ◽  
Powerful Mathematical Tool

This paper applies He’s parameter-expansion method to determine the limit cycle of oscillators in a u3/(1+u2) potential. The results are compared with the exact solutions. This shows that the method is a convenient and powerful mathematical tool for the search of limit cycles of nonlinear oscillators.

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