In this paper, we study the dynamics of a two-dimensional stochastic Navier–Stokes equation on a smooth domain, driven by linear multiplicative white noise. We show that solutions of the 2
D ...Navier–Stokes equation generate a perfect and locally compacting
C
1
,
1
cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium/stationary solution. We give sufficient conditions on the parameters of the Navier–Stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space.
We establish the existence of invariant stable manifolds for
C
1
perturbations of a nonuniform exponential dichotomy with an arbitrary nonuniform part. We consider the general case of sequences of ...maps, which corresponds to a nonautonomous dynamics with discrete time. We also obtain optimal estimates for the decay of trajectories along the stable manifolds. The optimal
C
1
smoothness of the invariant manifolds is obtained using an invariant family of cones.
For impulsive differential equations, we establish the existence of invariant stable manifolds under sufficiently small perturbations of a linear equation. We consider the general case of ...nonautonomous equations for which the linear part has a
nonuniform
exponential dichotomy. One of the main advantages of our work is that our results are optimal, in the sense that for vector fields of class
C
1
outside the jumping times, we show that the invariant manifolds are also of class
C
1
outside these times. The novelty of our proof is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, using the same approach we can also consider linear perturbations.
The Earth–Moon L1 libration point is proposed as a human gateway for space transportation system of the future. This paper studies indirect transfer using the perturbed stable manifold and lunar ...flyby to the Earth–Moon L1 libration point. Although traditional studies indicate that indirect transfer to the Earth–Moon L1 libration point does not save much fuel, this study shows that energy efficient indirect transfer using the perturbed stable manifold and lunar flyby could be constructed in an elegant way. The design process is given to construct indirect transfer to the Earth–Moon L1 libration point. Simulation results show that indirect transfer to the Earth–Moon L1 libration point saves about 420 m/s maneuver velocity compared to direct transfer, although the flight time is about 20 days longer.
We study asymptotic dynamics of the classical Lotka–Volterra competition model when both competing populations are subject to Allee effects. The resulting system can have up to four interior steady ...states. In such case, it is proved that both competing populations may either go extinct, coexist, or one population drives the other population to extinction depending on initial conditions.
For a nonautonomous linear equation
v
′
=
A
(
t
)
v
in a Banach space with a
nonuniform exponential dichotomy, we show that the nonlinear equation
v
′
=
A
(
t
)
v
+
f
(
t
,
v
,
λ
)
has stable ...invariant manifolds
V
λ
which are Lipschitz in the parameter
λ provided that
f is a sufficiently small Lipschitz perturbation. Since any linear equation with nonzero Lyapunov exponents has a nonuniform exponential dichotomy, the above assumption is very general. We emphasize that passing from a classical uniform exponential dichotomy to a general nonuniform exponential dichotomy requires a substantially new approach.
The hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set. A familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable ...manifolds at points on the chaotic set. Here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible (respectively noninvertible) maps of dimension 3 (respectively 2) and higher. In particular, we investigate a situation (first considered by Abraham and Smale in 1970 for different purposes) in which the dimension of the unstable (and stable) tangent spaces are not constant over the chaotic set; we call this
unstable dimension variability. A simple two-dimensional map that displays behavior typical of this phenomenon is presented and analyzed.
We study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space ℙ
2
.... We consider the perturbation (
z
2
+
ɛz
+
bɛw
2
,
w
2
) of (
z
2
,
w
2
) and we prove that, for
b
sufficiently small, it is injective on its basic set Λ
ɛ
close to Λ:= {0} ×
S
1
. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λ
ɛ
, in the case of these maps.
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