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In current mathematical literature, the perturbation theory in Hamiltonian systems has been established within the Hamiltonian framework, i.e., the perturbed system remains Hamiltonian. In the present model, the perturbation is dissipative. As far as we know, no previous study has investigated such a problem rigorously. The mathematical analysis of the present model by the authors will be published elsewhere. The basic idea is similar to the center manifold reduction: reducing a high-dimensional dynamical system into a lower-dimensional form. In Ref., singular perturbation method is employed to approximate the center manifold by assuming the existence of such a manifold. Our mathematical analysis proves its existence.
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In current mathematical literature, the perturbation theory in Hamiltonian systems has been established within the Hamiltonian framework, i.e., the perturbed system remains Hamiltonian. In the present model, the perturbation is dissipative. As far as we know, no previous study has investigated such a problem rigorously. The mathematical analysis of the present model by the authors will be published elsewhere. The basic idea is similar to the center manifold reduction: reducing a high-dimensional dynamical system into a lower-dimensional form. In Ref., singular perturbation method is employed to approximate the center manifold by assuming the existence of such a manifold. Our mathematical analysis proves its existence.
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See,. In this paper, the authors have shown that a chemical reaction system with every individual reaction being reversible can exhibit oscillatory dynamics. This is in complete agreement with the present study. However, the previous authors identified the reversible system with "near equilibrium." Even though it is mainly semantics, we believe that their terminology is misleading. As we have shown in the present paper, a reversible system can be either near or far from equilibrium depending on how severe the chemical driving force is.
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See D. Walz and S. R. Caplan, Biophys. J. 69, 1698 (1995). In this paper, the authors have shown that a chemical reaction system with every individual reaction being reversible can exhibit oscillatory dynamics. This is in complete agreement with the present study. However, the previous authors identified the reversible system with "near equilibrium." Even though it is mainly semantics, we believe that their terminology is misleading. As we have shown in the present paper, a reversible system can be either near or far from equilibrium depending on how severe the chemical driving force is.
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Strictly speaking, the LV system is a anharmonic oscillation since the frequencies of different periodic orbits vary in the system. In mechanics, harmonic oscillators are usually referred to linear oscillation whose frequencies are the same irrespective of the amplitudes of the oscillation. In the reversible LV system, not only the amplitudes of oscillation change but also its frequencies. The unperturbed Hamiltonian is nonlinear and frequencies for periodic orbits lying in different energy levels are not the same. Even though nonlinear transformations exist between the traditional LV system and a harmonic oscillator (e.g., Refs.), it is the nonlinear aspect of the LV system that gives rise to the transition between near to far from equilibrium.
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Strictly speaking, the LV system is a anharmonic oscillation since the frequencies of different periodic orbits vary in the system. In mechanics, harmonic oscillators are usually referred to linear oscillation whose frequencies are the same irrespective of the amplitudes of the oscillation. In the reversible LV system, not only the amplitudes of oscillation change but also its frequencies. The unperturbed Hamiltonian is nonlinear and frequencies for periodic orbits lying in different energy levels are not the same. Even though nonlinear transformations exist between the traditional LV system and a harmonic oscillator (e.g., Refs.), it is the nonlinear aspect of the LV system that gives rise to the transition between near to far from equilibrium.
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