Normal form computation without central manifold reduction

Leung, A.Y.T.; Zhang, Q.C.

doi:10.1016/s0022-460x(02)01626-7

Cited by 20 publications

(19 citation statements)

References 14 publications

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“…In this subsection, we investigate how much the dimension increases in the case of multiexponentially decaying kernels. In the case of uniform friction expressed by the sum of exponentials, the equation is reduced to polynomials, for which we definitely know the number of solutions for the nonlinear eigenvalue equation (15). Suppose the friction kernel has the following form:…”

Section: Dimensionality Of the Extended Systemmentioning

confidence: 99%

“…Recently, in order to extract the reaction coordinate and the no-return transition state for nonlinearly coupled multimode systems in a fluctuating environment, we have presented a theory [11][12][13][14] based on the concept of normal form 15 with the time-dependent formulation 16 within the framework of the multidimensional Langevin formulation. It was shown that, under certain conditions, a nonlinear coordinate transformation can be performed to provide a new reaction coordinate independent of all the other coordinates similarly to the case of Hamiltonian systems in the region of saddles.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic reaction coordinate in thermally fluctuating environment in the framework of the multidimensional generalized Langevin equations

Kawai

Komatsuzaki

2010

Phys. Chem. Chem. Phys.

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Nonlinear generalized Langevin equation with memory due to thermal environment is equivalent to a memoryless equation with increased dimensionality A framework recently developed for the extraction of a dynamic reaction coordinate to mediate reactions buried in multidimensional Langevin equation is extended to the generalized Langevin equations without a priori assumption on the forms of the potential (in general, nonlinearly coupled systems) and the friction kernel. The equation of motion with memory effect can be transformed into an equation without memory at the cost of an increase in the dimensionality of the system, and hence the theoretical framework developed for the (nonlinear) Langevin formulation can be generalized to the non-Markovian process with colored noise. It is found that the increased dimension can be physically interpreted as effective modes of the fluctuating environment. As an illustrative example, we apply this theory to a multidimensional generalized Langevin equation for motion on the Müller-Brown potential surface with an exponential friction kernel. Numerical simulations find a boundary between the highly reactive region and the less reactive region in the space of initial conditions. The location of the boundary is found to depend significantly on both the memory kernel and the nonlinear couplings. The theory extracts a reaction coordinate whose sign determines the fate of the reaction taking into account the thermally fluctuating environments, the memory effect, and the nonlinearities. It is found that the location of the boundary of reactivity is satisfactorily reproduced as the zero of the statistical average of the new reaction coordinate, which is an analytical functional of both the original position coordinates and velocities of the system, and of the properties of the environment.

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Section: Dimensionality Of the Extended Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamic reaction coordinate in thermally fluctuating environment in the framework of the multidimensional generalized Langevin equations

Kawai

Komatsuzaki

2010

Phys. Chem. Chem. Phys.

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“…Normal form transformations are a powerful technique for studying the response of nonlinear oscillators, which has it's origins in the work of Poincaré [7]. The techniques have a long history of development and application, and are particularly useful for identifying resonant interactions in oscillators -for example see [8,9,10,11,12,13] and the related approach of using nonlinear normal modes [14,15,16,17,18]. In addition, a comprehensive overview of normal form theory and related techniques can be found in [19,20,21,22], and a survey of recent developments is given by Stolovitch [23].…”

Section: Introductionmentioning

confidence: 99%

Resonant response functions for nonlinear oscillators with polynomial type nonlinearities

Xin

Neild

Wagg

et al. 2013

Journal of Sound and Vibration

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In this paper we consider the steady-state response of forced, damped, weakly nonlinear oscillators with polynomial type nonlinearities. In particular we define general expressions that can be used to compute resonant response functions which define the steady state constant amplitude oscillatory response at the primary resonance and the associated harmonics. The resonant response functions are derived using a normal form transformation which is carried out directly on the second order nonlinear oscillator. The example of a forced Van der Pol oscillator with an additional cubic stiffness nonlinearity is used to demonstrate how the general analysis can be applied.

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“…The term "underdamped" means that the second time derivative of the position q is included in the equation of motion, which is distinguished from the "overdamped" case of q = 0, where equilibration in the velocity ͑q ͒ space is supposed to be attained much more quickly than in the position ͑q͒ space. The crux is the application of the non-Hamiltonian NF theory 54 together with the time-dependent formulation given by Ref. 55.…”

Section: Introductionmentioning

confidence: 99%

Dynamic pathways to mediate reactions buried in thermal fluctuations. I. Time-dependent normal form theory for multidimensional Langevin equation

Kawai

Komatsuzaki

2009

The Journal of Chemical Physics

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We present a novel theory which enables us to explore the mechanism of reaction selectivity and robust functions in complex systems persisting under thermal fluctuation. The theory constructs a nonlinear coordinate transformation so that the equation of motion for the new reaction coordinate is independent of the other nonreactive coordinates in the presence of thermal fluctuation. In this article we suppose that reacting systems subject to thermal noise are described by a multidimensional Langevin equation without a priori assumption for the form of potential. The reaction coordinate is composed not only of all the coordinates and velocities associated with the system ͑solute͒ but also of the random force exerted by the environment ͑solvent͒ with friction constants. The sign of the reaction coordinate at any instantaneous moment in the region of a saddle determines the fate of the reaction, i.e., whether the reaction will proceed through to the products or go back to the reactants. By assuming the statistical properties of the random force, one can know a priori a well-defined boundary of the reaction which separates the full position-velocity space in the saddle region into mainly reactive and mainly nonreactive regions even under thermal fluctuation. The analytical expression of the reaction coordinate provides the firm foundation on the mechanism of how and why reaction proceeds in thermal fluctuating environments.

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Normal form computation without central manifold reduction

Cited by 20 publications

References 14 publications

Dynamic reaction coordinate in thermally fluctuating environment in the framework of the multidimensional generalized Langevin equations

Dynamic reaction coordinate in thermally fluctuating environment in the framework of the multidimensional generalized Langevin equations

Resonant response functions for nonlinear oscillators with polynomial type nonlinearities

Dynamic pathways to mediate reactions buried in thermal fluctuations. I. Time-dependent normal form theory for multidimensional Langevin equation

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