scholarly journals Center Manifolds for Infinite Dimensional Nonautonomous Differential Equations

1997 ◽  
Vol 141 (2) ◽  
pp. 356-399 ◽  
Author(s):  
C. Chicone ◽  
Y. Latushkin
Keyword(s):  
Differential Equations ◽  
Center Manifolds ◽  
Infinite Dimensional ◽  
Nonautonomous Differential Equations

scholarly journals The Bohl Spectrum for Linear Nonautonomous Differential Equations

2016 ◽  
Vol 29 (4) ◽  
pp. 1459-1485 ◽  
Author(s):  
Thai Son Doan ◽  
Kenneth J. Palmer ◽  
Martin Rasmussen
Keyword(s):  
Differential Equations ◽  
Nonautonomous Differential Equations

scholarly journals MORSE SPECTRUM FOR NONAUTONOMOUS DIFFERENTIAL EQUATIONS

2008 ◽  
Vol 08 (03) ◽  
pp. 351-363 ◽  
Author(s):  
FRITZ COLONIUS ◽  
PETER E. KLOEDEN ◽  
MARTIN RASMUSSEN
Keyword(s):  
Differential Equations ◽  
Finite Time ◽  
Accumulation Point ◽  
Morse Decomposition ◽  
Finite Time Lyapunov Exponents ◽  
The Past ◽  
Accumulation Points ◽  
Morse Spectrum ◽  
Nonautonomous Differential Equations ◽  
Linear Cocycle

The concept of a Morse decomposition consisting of nonautonomous sets is reviewed for linear cocycle mappings w.r.t. the past, future and all-time convergences. In each case, the set of accumulation points of the finite-time Lyapunov exponents corresponding to points in a nonautonomous set is shown to be an interval. For a finest Morse decomposition, the Morse spectrum is defined as the union of all of the above accumulation point intervals over the different nonautonomous sets in such a finest Morse decomposition. In addition, Morse spectrum is shown to be independent of which finest Morse decomposition is used, when more than one exists.

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

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