This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial ...approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton--Kantorovich argument in order to obtain a kind of "analytic-shadowing" result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six. PUBLICATION ABSTRACT
We investigate asymptotic dynamics of the classical Leslie-Gower competition model when both competing populations are subject to Allee effects. The system may possess four interior steady states. It ...is proved that for certain parameter regimes both competing populations may either go extinct, coexist or one population drives the other population to extinction depending on initial conditions.
Consider an evolution family
U
=
(
U
(
t
,
s
)
)
t
⩾
s
⩾
0
on a half-line
R
+
and a semi-linear integral equation
u
(
t
)
=
U
(
t
,
s
)
u
(
s
)
+
∫
s
t
U
(
t
,
ξ
)
f
(
ξ
,
u
(
ξ
)
)
d
ξ
. We prove ...the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of
L
p
type, the Lorentz spaces
L
p
,
q
and many other function spaces occurring in interpolation theory. The existence of such manifolds is obtained in the case that
(
U
(
t
,
s
)
)
t
⩾
s
⩾
0
has an exponential dichotomy and the nonlinear forcing term
f
(
t
,
x
)
satisfies the non-uniform Lipschitz conditions:
‖
f
(
t
,
x
1
)
−
f
(
t
,
x
2
)
‖
⩽
φ
(
t
)
‖
x
1
−
x
2
‖
for
φ being a real and positive function which belongs to certain classes of admissible function spaces.
We study a recently proposed somatotroph model that exhibits plateau bursting, a form of electrical activity that is typical for this cell type. We focus on the influence of the large conductance ...(BK-type)
Ca
2
+
-activated
K
+
current on the oscillations and duration of the active phase. The model involves two different time scales, but a standard bifurcation analysis of the fast time limit does not completely explain the behaviour of the model, which is subtly different from classical models for plateau bursting. In particular, the nullclines and velocities of the fast variables play an important role in shaping the bursting oscillations. We determine numerically how the fraction of open BK channels controls the amplitude of the fast oscillations during the active phase. Furthermore, we show how manifolds of the fast subsystem are involved in the termination of the active phase.
Under some appropriate conditions, we prove the existence and uniqueness of periodic solutions to partial functional differential equations with infinite delay of the form
u
̇
=
A
(
t
)
u
+
g
(
t
,
u
...t
)
on a Banach space
X
where
A
(
t
) is 1-periodic, and the nonlinear term
g
(
t
,
ϕ
) is 1-periodic with respect to
t
for each fixed
ϕ
in fading memory phase spaces, and is
φ
(
t
)-Lipschitz for
φ
belonging to an admissible function space. We then apply the attained results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family (
A
(
t
))
t
≥ 0
generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.
In this paper, we investigate local stable manifold for planar Hadamard fractional differential equations. By adopting Lyapunov–Perron operator approach and establishing new estimation of ...Mittag-Leffler function associated with Hadamard fractional derivative, we derive another interesting local stable manifold theorem for our problem. Finally, an example is given to illustrate our theoretical results.
We study the stability under perturbations for delay difference equations in Banach spaces. Namely, we establish the (nonuniform) stability of linear
nonuniform exponential contractions under ...sufficiently small perturbations. We also obtain a stable manifold theorem for perturbations of linear delay difference equations admitting a nonuniform exponential dichotomy, and show that the stable manifolds are Lipschitz in the perturbation.
The Real Dynamics of Bieberbach’s Example Hayes, Sandra; Hundemer, Axel; Milliken, Evan ...
The Journal of geometric analysis,
10/2015, Letnik:
25, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Bieberbach constructed, in 1933, domains in
C
2
which were biholomorphic to
C
2
but not dense. The existence of such domains was unexpected. The special domains Bieberbach considered are basins of ...attraction of a cubic Hénon map. This classical method of construction is one of the first applications of dynamical systems to complex analysis. In this paper, the boundaries of the real sections of Bieberbach’s domains will be calculated explicitly as the stable manifolds of the saddle points. The real filled Julia sets and the real Julia sets of Bieberbach’s map will also be calculated explicitly and illustrated with computer generated graphics. Basic differences between real and the complex dynamics will be shown.
In this paper, we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important centre ...manifolds, and study their bifurcation. Saddle-node and period-doubling bifurcation route to chaos are exhibited via numerical simulations.
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