Periodic and Steady-State Mode Interactions Lead to Tori

Langford, William F.

doi:10.1137/0137003

Cited by 153 publications

(57 citation statements)

References 32 publications

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“…Nonlinear systems with periodic driving forces have been considered in detail by, for example, Nayfeh and Mook [1], Iooss and Joseph [2], who used perturbation techniques and operator function theory to obtain the first-order approximations of bifurcation solutions and stability conditions. In a nonlinear autonomous system, interactions of static and dynamic bifurcation modes in the vicinity of a compound critical point may lead to invariant tori, which has been investigated by many authors (e.g., see [3][4][5][6][7]). Recently, such phenomena are studied by using the unification technique [7][8][9][10][11] coupled with the intrinsic harmonic balancing [12] and its generalization, multiple-scale intrinsic harmonic balancing [11].…”

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confidence: 99%

On phase-locked motions associated with strong resonance

Yu¹,

Huseyin²

1993

Quart. Appl. Math.

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Abstract. This paper is concerned with the stability and bifurcation behaviour of a nonlinear autonomous system in the vicinity of a compound critical point characterized by two pairs of pure imaginary eigenvalues of the Jacobian. Attention is focused on the local dynamics of the system near-to-resonance. The methodology developed earlier for the bifurcation analysis into periodip and quasi-periodic motions (unification technique coupled with the intrinsic harmonic balancing) is extended to consider the stability and bifurcations of resonant cases. A set of simplified rate equations characterizing the local dynamics of the system is derived. These equations differ from those associated with nonresonant cases in that they are phase-coupled. Furthermore, the stability conditions of the phase-locked periodic bifurcation solutions are presented. All the results are expressed in explicit forms.1. Introduction. It is well known that a nonautonomous system may quite often exhibit periodic as well as quasi-periodic bifurcation solutions. Nonlinear systems with periodic driving forces have been considered in detail by, for example, Nayfeh and Mook [1], Iooss and Joseph [2], who used perturbation techniques and operator function theory to obtain the first-order approximations of bifurcation solutions and stability conditions. In a nonlinear autonomous system, interactions of static and dynamic bifurcation modes in the vicinity of a compound critical point may lead to invariant tori, which has been investigated by many authors (e.g., see [3][4][5][6][7]). Recently, such phenomena are studied by using the unification technique [7][8][9][10][11] coupled with the intrinsic harmonic balancing [12] and its generalization, multiple-scale intrinsic harmonic balancing [11]. The approach enables one to obtain systematically a set of simplified rate equations which govern the local dynamics of the system. The solutions for static bifurcations, Hopf bifurcations, and quasi-periodic motions lying on two-and higher-dimensional invariant tori, as well as the associated stability conditions, are expressed in terms of the system coefficients.

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confidence: 99%

On phase-locked motions associated with strong resonance

Yu¹,

Huseyin²

1993

Quart. Appl. Math.

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“…Let us show that equation (5.9) undergoes both a Hopf bifurcation and steady state bifurcation asσ passes through zero. Motivated by [8,12], we expect that interactions between the two modes can lead to secondary bifurcations. Although later we shall specialize to the values of m and n in (5.12), any values satisfying n > 1, mn < 1 yield qualitatively similar behavior (see Chap.…”

Section: Analysis Of the Bifurcationmentioning

confidence: 99%

Bifurcations in a modulation equation for alternans in a cardiac fiber

Dai

Schaeffer

2010

ESAIM: M2AN

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Abstract. While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. As shown in our previous paper, the zero solution of their equation may lose stability, as the pacing rate is increased, through either a Hopf or steady-state bifurcation. Which bifurcation occurs first depends on parameters in the equation, and for one critical case both modes bifurcate together at a degenerate (codimension 2) bifurcation. For parameters close to the degenerate case, we investigate the competition between modes, both numerically and analytically. We find that at sufficiently rapid pacing (but assuming a 1:1 response is maintained), steady patterns always emerge as the only stable solution. However, in the parameter range where Hopf bifurcation occurs first, the evolution from periodic solution (just after the bifurcation) to the eventual standing wave solution occurs through an interesting series of secondary bifurcations.Mathematics Subject Classification. 35B32, 92C30.

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“…The analysis will be made by reducing the problem to a three-dimensional center manifold and obtaining its normal form; then we use the general analysis of this normal form in [9]. Let us mention that similar (but different) normal forms are obtained at the interaction of a Hopf bifurcation point and either a transcritical or a pitchfork steady bifurcation point; see [26], [27] and also [9, pp. 396 and 410].…”

Section: 2mentioning

confidence: 99%

Weakly Nonuniform Thermal Effects in a Porous Catalyst: Asymptotic Models and Local Nonlinear Stability of the Steady States

Mancebo¹,

Vega²

1992

SIAM J. Appl. Math.

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Abstract. This paper considers a first-order, irreversible exothermic reaction in a bounded porous catalyst, with smooth boundary, in one, two, and three space dimensions. It is assumed that the characteristic reaction time is sufficiently small for the chemical reaction to be confined to a thin layer near the boundary of the catalyst, and that the thermal diffusivity is large enough for the temperature to be uniform in the reaction layer, but that it is not so large as to avoid significant thermal gradients inside the catalyst. For appropriate realistic limiting valúes of the several nondimensional parameters of the problem, severa! time-dependent asymptotic models are derived that account for the chemícal reaction at the boundary (that becomes essentially impervious to the reactant), heat conduction inside the catalyst, and exchange of heat and reactant with the surrounding unreacted fluid, These models possess asymmetrical steady states for symmetric shapes of the catalyst, and some of them exhibit a rich dynamic behavior that includes quasiperiodic phenomena. In one case, the linear stability of the steady states, and also the local bifurcatíon to quasi-periodic solutions via center manifold theory and normal form reduction, are analyzed.Key words, porous catalysts, weakly nonlinear stability, normal forms AMS(MOS) subject classifications. 35B32, 35KS7, 80A30, 80A32I. Introductiort and formulation. This paper deals with a well-known model for the evolution of the reactant concentration u and of the temperature v in a porous catalyst, in which a first-order, irreversible, exothermic reaction occurs. After suitable nondimensionalization (length is referred to a characteristic dimensión of the catalyst, time is referred to the thermal diffusion time within the catalyst, and the reactant concentration and temperature are referred to their respective valúes at the external unreacted fluid), the principie of conservaron applied to the reactant and to enthalpy leads to the following model [1, Vol. I]:for t >0, with appropriate initiai conditions at í -0. Here A is the Laplacian operator, n is the outward unit normal to the smooth boundary of the domain (1<=W (p -1, 2, or 3), and all the parameters are positive. 2 (Damkohler number) is the ratio of the reaction rate to the difíusion rate, y is the acüvathn energy (or temperature) of the chemical reaction, L (Lewis number) is the ratio of thermal to material diífusivity, 0 (Prater number) is a measure of the chemicai heat reiease (/3L is a measure of the heat of reaction relative to the thermal energy of the catalyst), and a-and v (diffusional and thermal Biot numbers) are measures of the rates of mass and heat transfer between the surface of catalyst and the external fluid, relative to the rates of mass and heat transfer within the cataiyst.Usually, the thermal diffusivity of the soiid (metallicorwithmetalliccomponents) catalyst is very high, and, consequently, ¡3 is quite small, v is fairly small or of order

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Periodic and Steady-State Mode Interactions Lead to Tori

Cited by 153 publications

References 32 publications

On phase-locked motions associated with strong resonance

On phase-locked motions associated with strong resonance

Bifurcations in a modulation equation for alternans in a cardiac fiber

Weakly Nonuniform Thermal Effects in a Porous Catalyst: Asymptotic Models and Local Nonlinear Stability of the Steady States

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