Lyapunov Exponents of the Half-Line SHE

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

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Asymptotic behavior of solutions of periodic linear partial functional differential equations on the half line

Journal of Differential Equations ◽

10.1016/j.jde.2021.05.053 ◽

2021 ◽

Vol 296 ◽

pp. 1-14

Author(s):

Vu Trong Luong ◽

Nguyen Huu Tri ◽

Nguyen Van Minh

Keyword(s):

Asymptotic Behavior ◽

Differential Equations ◽

Functional Differential Equations ◽

Half Line ◽

Asymptotic Behavior Of Solutions ◽

Partial Functional Differential Equations ◽

Periodic Linear ◽

Functional Differential

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Linear evolution equations on the half-line with dynamic boundary conditions

European Journal of Applied Mathematics ◽

10.1017/s0956792521000103 ◽

2021 ◽

pp. 1-33

Author(s):

D. A. SMITH ◽

W. Y. TOH

Keyword(s):

Boundary Conditions ◽

Heat Equation ◽

Evolution Equations ◽

Half Line ◽

Robin Problem ◽

Dynamic Boundary Conditions ◽

Transform Method ◽

Linear Evolution ◽

Robin Condition ◽

Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

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