文档介绍：FINITE DIMENSIONAL REDUCTION
OF NONAUTONOMOUS DISSIPATIVE
SYSTEMS
Alain Miranville
Universit´ede Poitiers
Collaborators : M. Efendiev and S. Zelik
Long time behavior of equations of the form
y′= F (t, y)
For autonomous systems :
y′= F (y)
In many situations, the evolution of the sys-
tem is described by a system of ODEs :
N
y = (y1, ..., yN ) ∈ R , F = (F1, ..., FN )
Assuming that the Cauchy problem
y′= F (y),
y(0) = y0,
is well-posed, we can define the family of solv-
ing operators S(t), t ≥ 0, acting on a subset
φ⊂ RN :
S(t) : φ→φ
y0 7→ y(t)
This family of operators satisfies
S(0) = Id,
S(t + s) = S(t) ◦ S(s), t, s ≥ 0
We say that it forms a semigroup on φ
Qualitative study of such systems : goes back
to Poincar´e
Much is known nowadays, at least in low di-
mensions
Even relatively simple systems can generate
plicated chaotic behaviors
These systems are sensitive to perturbations :
trajectories with close initial data may diverge
exponentially
→ Temporal evolution unpredictable on ti-
me scales larger than some critical value
→ Show typical stochastic behaviors
Example : Lorenz system
x′= σ(y − x)
y′= −xy + rx − y
z′= xy − bz
Obtained by truncature of the Navier-Stokes
equations
Gives an approximate description of a layer of
fluid heated from below
Similar to what is observed in the atmosphere
For a sufficiently intense heating : sensitive
dependence on the initial conditions, repre-
sents a very irregular convection
→ Butterfly effect
Very often, the trajectories are localized in
some subset of the phase space having a very
complicated geometric structure (., locally
homeomorphic to the product of Rm and a
Cantor set)
→ Strange attractor (Ruelle and Takens)
Main feature of a strange attractor : dimen-
sion
Sensitivity to initial conditions : > 2 (dimen-
sion of the phase space ≥ 3, say, 3)
Contraction of volumes : its volume is equal
to 0
→ noninteger, strictly between 2 and 3
→ Fra