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### Open Access Journals

IPI

This paper is devoted to exploring the effects of non-Gaussian
fluctuations on dynamical evolution of a tumor growth model with
immunization, subject to non-Gaussian $\alpha$-stable type Lévy
noise. The corresponding deterministic model has two meaningful
states which represent the state of tumor extinction and the state
of stable tumor, respectively. To characterize the time for
different initial densities of tumor cells staying in the domain
between these two states and the likelihood of crossing this domain,
the mean exit time and the escape probability are quantified by
numerically solving differential-integral equations with appropriate
exterior boundary conditions. The relationships between the
dynamical properties and the noise parameters are examined. It is
found that in the different stages of tumor, the noise parameters
have different influences on the time and the likelihood inducing
tumor extinction. These results are relevant for determining
efficient therapeutic regimes to induce the extinction of tumor
cells.

DCDS-B

The Thermohaline Circulation, which plays a crucial role in
the global climate, is a cycle of deep ocean due to
the change in salinity and temperature (i.e., density). The effects of non-Gaussian noise on
the Stommel box model for the Thermohaline Circulation are considered. The noise
is represented by a non-Gaussian $\alpha$-stable Lévy motion with $0<\alpha < 2$. The $\alpha$ value may be regarded as the index of non-Gaussianity. When $\alpha=2$, the $\alpha$-stable Lévy motion becomes the usual (Gaussian) Brownian motion.

Dynamical features of this stochastic model is examined by computing the mean exit time for various $\alpha$ values. The mean exit time is simulated by numerically solving a deterministic differential equation with nonlocal interactions. It has been observed that some salinity difference levels remain in certain ranges for longer times than other salinity difference levels, for different $\alpha$ values. This indicates a lower variability for these salinity difference levels. Realizing that it is the salinity differences that drive the thermohaline circulation, this lower variability could mean a stable circulation, which may have further implications for the global climate dynamics.

Dynamical features of this stochastic model is examined by computing the mean exit time for various $\alpha$ values. The mean exit time is simulated by numerically solving a deterministic differential equation with nonlocal interactions. It has been observed that some salinity difference levels remain in certain ranges for longer times than other salinity difference levels, for different $\alpha$ values. This indicates a lower variability for these salinity difference levels. Realizing that it is the salinity differences that drive the thermohaline circulation, this lower variability could mean a stable circulation, which may have further implications for the global climate dynamics.

DCDS-B

Stochastic averaging for a class of stochastic differential
equations (SDEs) with fractional Brownian motion, of the Hurst parameter $H$ in
the interval $(\frac{1}{2},1)$, is investigated. An averaged SDE for the
original SDE is proposed, and their solutions are quantitatively compared. It
is shown that the solution of the averaged SDE converges to that of the
original SDE in the sense of mean square and also in probability. It is further
demonstrated that a similar averaging principle holds for SDEs under
stochastic integral of pathwise backward and forward types. Two examples are
presented and numerical simulations are carried
out to illustrate the averaging principle.

DCDS-B

The ocean thermohaline circulation, also called meridional overturning
circulation, is caused by water density contrasts. This circulation has
large capacity of carrying heat around the globe and it thus affects the energy
budget and further affects the climate. We consider a thermohaline circulation
model in the meridional plane under external wind forcing. We show that,
when there is no wind forcing, the stream function and the density fluctuation
(under appropriate metrics) tend to zero exponentially fast as time goes to
infinity. With rapidly oscillating wind forcing, we obtain an averaging principle
for the thermohaline circulation model. This averaging principle provides
convergence results and comparison estimates between the original thermohaline
circulation and the averaged thermohaline circulation, where the wind
forcing is replaced by its time average. This establishes the validity for using
the averaged thermohaline circulation model for numerical simulations at long
time scales.

DCDS-B

The usual Wong-Zakai approximation is about simulating individual solutions
of stochastic differential equations(SDEs). From the perspective of dynamical systems, it is also interesting to approximate random invariant
manifolds which are geometric objects useful for understanding how complex dynamics evolve under stochastic
influences. We study a Wong-Zakai type of approximation for the random stable manifold of a stochastic evolutionary equation with multiplicative correlated noise. Based on the convergence of solutions on the
invariant manifold, we approximate the random stable manifold by the invariant manifolds of a family of perturbed stochastic
systems with smooth correlated noise (i.e., an integrated Ornstein-Uhlenbeck process). The convergence of this approximation is established.

DCDS-B

The dynamical behavior of non-autonomous strongly damped wave-type evolutionary equations with linear memory, critical nonlinearity, and time-dependent external forcing is investigated. The time-dependent external forcing is assumed to be only translation-bounded, instead of translation-compact. First, the asymptotic regularity of solutions is proved, and then the existence of the compact uniform attractor together with its structure and regularity is obtained.

DCDS-S

Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense.
Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.

DCDS-B

This work is devoted to investigate the well-posedness and long-time behavior of solutions for the following nonlocal nonlinear partial differential equations in a bounded domain

$\begin{align*}u_t+(-Δ)^{σ/2}u +f(u) = g.\end{align*}$ |

Firstly, due to the lack of an upper growth restriction of the nonlinearity $f$, we have to utilize a weak compactness approach in an Orlicz space to obtain the well-posedness of weak solutions for the equations. We then establish the existence of

-absorbing sets and

-absorbing sets for the solution semigroup

. Finally, we prove the existence of

-global attractor and

-global attractor by a asymptotic compactness method.

$(L^2_0(Ω), L^2_0(Ω))$ |

$(L^2_0(Ω), H^{σ/2}_0(Ω))$ |

$\{S(t)\}_{t≥q 0}$ |

$(L^2_0(Ω), L^2_0(Ω))$ |

$(L^2_0(Ω), H^{σ/2}_0(Ω))$ |

DCDS-B

We prove the existence of recurrent or Poisson stable
motions in the Navier-Stokes
fluid system under recurrent or Poisson stable forcing, respectively.
We use an approach
based on nonautonomous dynamical systems ideas.

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