Analysis of nonlinear modes of variation for functional data
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Variation (astronomy)
The dynamics of a disordered nonlinear chain can be either regular or chaotic with a certain probability. The chaotic behavior is often associated with the destruction of Anderson localization by the nonlinearity. In the presentwork it is argued that at weak nonlinearity chaos is nucleated locally on rare resonant segments of the chain. Based on this picture, the probability of chaos is evaluated analytically. The same probability is also evaluated by direct numerical sampling of disorder realizations and quantitative agreement between the two results is found.
Chain (unit)
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In fluid flows, fluid-structure interactions and other fluids-related problem, nonlinear dynamics play an important role in determining the development, response or output. Understanding these dynamics is essential for development of analytical models and prediction and control purposes. Higher-order statistical analysis has been shown to be an effective tool that can be applied to identify nonlinear couplings and measure energy transfer rates. These techniques have been applied by our group and others to investigate transition of shear flows, energy cascading in turbulence, oceanographic and geophysical flows, and fluid-structure interactions. The results of these investigations revealed important nonlinear characteristics regarding these problems. In this paper, we review these techniques and explain their usefulness in identification and quantification of nonlinear dynamics.
Lagrangian Coherent Structures
Geophysical fluid dynamics
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In this paper,firstly,we introduce the development history of constant variation method and averaging method.Then, we study the averaging method and its relationship with the constant variation method in solving the weakly nonlinear system +ω_0～2=ef(t,x,).Eventually,we give an example,which illustrates the effectiveness of the averaging method.
Variation (astronomy)
Constant (computer programming)
Method of averaging
Variation of parameters
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Caratheodory’s theorem on small witnesses for convex hulls of sets is shown to have a natural analogue for finitely supported measures. Contrast is drawn with the much larger witnesses required for multisets, as shown by Barany and Perles.
Variation (astronomy)
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Complex system
Solid-state physics
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Thermal fluctuations
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We study order–disorder transitions and the emergence of collective behavior using a particular mean field model: the dynamic Takatsuji system. This model satisfies linear non-equilibrium thermodynamics and can be described in terms of a nonlinear Markov process defined by a nonlinear Fokker–Planck equation, that is, an evolution equation that is nonlinear with respect to its probability density. We discuss quantitatively the impact of a feedback loop that involves a macroscopic, thermodynamic variable. We demonstrate by means of semi-analytical methods and numerical simulations that the feedback loop increases the magnitude of order, increases the gap between the free energy of the ordered and disordered states, and increases the maximal rate of entropy production that can be observed during the order–disorder transition.
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We show, using detailed numerical analysis and theoretical arguments, that the normalized participation number of the stationary solutions of disordered nonlinear lattices obeys a one-parameter scaling law. Our approach opens a new way to investigate the interplay of Anderson localization and nonlinearity based on the powerful ideas of scaling theory.
Scaling law
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Statistical Mechanics
Statistical theory
Basis (linear algebra)
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