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1078-0947

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## Discrete & Continuous Dynamical Systems - A

2016 , Volume 36 , Issue 1

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2016, 36(1): 1-41
doi: 10.3934/dcds.2016.36.1

*+*[Abstract](405)*+*[PDF](620.8KB)**Abstract:**

We consider diffeomorphisms of compact Riemmanian manifolds which have a Gibbs-Markov-Young structure, consisting of a reference set $\Lambda$ with a hyperbolic product structure and a countable Markov partition. We assume polynomial contraction on stable leaves, polynomial backward contraction on unstable leaves, a bounded distortion property and a certain regularity of the stable foliation. We establish a control on the decay of correlations and large deviations of the physical measure of the dynamical system, based on a polynomial control on the Lebesgue measure of the tail of return times. Finally, we present an example of a dynamical system defined on the torus and prove that it verifies the properties of the Gibbs-Markov-Young structure that we considered.

2016, 36(1): 43-61
doi: 10.3934/dcds.2016.36.43

*+*[Abstract](256)*+*[PDF](836.0KB)**Abstract:**

Let $f$ be a continuous map from the unit interval to itself. In this paper, we investigate the structure of space $Z$ which is constructed corresponding to the behaviors of $f$ and a periodic orbit $P$ of $f$. Under some restriction of $f$, we get necessary and sufficient conditions for $Z$ being the universal dendrite. Furthermore $Z$ is classified into five types especially when it is a tree.

2016, 36(1): 63-95
doi: 10.3934/dcds.2016.36.63

*+*[Abstract](481)*+*[PDF](581.9KB)**Abstract:**

The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and establish a connection with an ancestral partitioning process, backward in time. We solve the finite-dimensional problem recursively for generic sets of parameters and briefly discuss the singular cases, and how to extend the solution to this situation.

2016, 36(1): 97-136
doi: 10.3934/dcds.2016.36.97

*+*[Abstract](458)*+*[PDF](589.4KB)**Abstract:**

In this paper, we improve the known estimates for the invariance entropy of a nonlinear control system. For sets of complete approximate controllability we derive an upper bound in terms of Lyapunov exponents and for uniformly hyperbolic sets we obtain a similar lower bound. Both estimates can be applied to hyperbolic chain control sets, and we prove that under mild assumptions they can be merged into a formula. The proof of our result reveals the interesting qualitative statement that there exists no control strategy to make a uniformly hyperbolic chain control set invariant that cannot be beaten or at least approached (in the sense of lowering the necessary data rate) by the strategy to stabilize the system at a periodic orbit in the interior of this set.

2016, 36(1): 137-149
doi: 10.3934/dcds.2016.36.137

*+*[Abstract](405)*+*[PDF](369.8KB)**Abstract:**

For state-dependent delay equations, it may easily happen that the equation is not differentiable. This hampers the formulation and the proof of the Principle of Linearized Stability. The fact that an equation is not differentiable does not, by itself, imply that the solution operators are not differentiable. And indeed, the aim of this paper is to present a simple example with differentiable solution operators despite of lack of differentiability of the equation. The example takes the form of a renewal equation and is motivated by a population dynamical model.

2016, 36(1): 151-169
doi: 10.3934/dcds.2016.36.151

*+*[Abstract](542)*+*[PDF](460.8KB)**Abstract:**

This paper deals with the chemotaxis system \[ \begin{cases} u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in \Omega, \ t>0, \\ v_t=\Delta v + wz, \qquad x\in \Omega, \ t>0, \\ w_t=-wz, \qquad x\in \Omega, \ t>0, \\ z_t=\Delta z - z + u, \qquad x\in \Omega, \ t>0, \end{cases} \] in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \le 3$, that has recently been proposed as a model for tumor invasion in which the role of an active extracellular matrix is accounted for.

It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a corresponding initial-boundary value problem of Neumann type possesses a global solution which is bounded. Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certain spatially homogeneous equilibrium in the sense that as $t\to\infty$,

$u(x,t)\to \overline{u_0}$ , $v(x,t) \to \overline{v_0} + \overline{w_0}$, $w(x,t) \to 0$ and $z(x,t) \to \overline{u_0}$, uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$, $\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$ and $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.

2016, 36(1): 171-215
doi: 10.3934/dcds.2016.36.171

*+*[Abstract](458)*+*[PDF](1832.4KB)**Abstract:**

This paper is the continuation of our previous papers [16] and [17] where we studied small-amplitude limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations. We find optimal upper bounds for the number of small-amplitude limit cycles in these slow-fast codimension 3 bifurcations. We use techniques from geometric singular perturbation theory.

2016, 36(1): 217-244
doi: 10.3934/dcds.2016.36.217

*+*[Abstract](420)*+*[PDF](118.9KB)**Abstract:**

We study the fixed point index on the carrying simplex of the competitive map. The sum of the indices of the fixed points on the carrying simplex for the three-dimensional competitive map is unit. Based on that, we analyze the asymptotic behavior of the three-dimensional competitive Atkinson/Allen model. We present all the equivalence classes relative to the boundary of the carrying simplex of the low-dimensional (two or three) map, depending upon relationship among the model coefficients. For the two-dimensional case, there are only three dynamic scenarios, and every orbit converges to some fixed point. For the three-dimensional case, there are total $33$ stable equivalence classes, and in $18$ of them all the compact limit sets are fixed points. Further, we focus on the analysis of the dynamics of the other $15$ cases. Hopf bifurcation is studied and a necessary condition for it occurring is given, which implies that the classes $19$-$25$, $28$, $30$ and $32$ do not have any Hopf bifurcation. However, the class $26$ and class $27$ do admit Hopf bifurcations, which means that these two classes may have isolated invariant closed curves in their carrying simplex, and such an invariant closed curve corresponds to either a subharmonic or a quasiperiodic solution in continuous time systems. Each system in class $27$ has a heteroclinic cycle and the numerical simulation also reveals that there exist systems having May-Leonard phenomenon: the existence of nonquasiperiodic oscillation.

2016, 36(1): 245-259
doi: 10.3934/dcds.2016.36.245

*+*[Abstract](290)*+*[PDF](378.5KB)**Abstract:**

We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Hölder regularity. We also study cohomology of cocycles and its connection with holonomies. We obtain a result on regularity of a measurable conjugacy, as well as a necessary and sufficient condition for existence of a continuous conjugacy between two cocycles.

2016, 36(1): 261-277
doi: 10.3934/dcds.2016.36.261

*+*[Abstract](318)*+*[PDF](502.8KB)**Abstract:**

In this paper we are concerned with the regularity of weak solutions $u$ to the one phase continuous casting problem $$ div (A(x) \nabla u(X)) = div [\beta (u) v(X)], X\in \mathcal{C}_L$$ in the cylindrical domain $\mathcal{C}_L=\Omega\times (0,L)$ where $X=(x,z), x\in \Omega\subset \mathbb{R}^{N-1}, z\in(0,L), L>0$ with given elliptic matrix $A:\Omega \to \mathbb{R}^{N^2}, A_{ij}(x)\in C^{1,\alpha_0}(\Omega), \alpha_0 > 0$, prescribed convection $v$, and the enthalpy function $\beta(u)$. We first establish the optimal regularity of weak solutions $u\ge 0$ for one phase problem. Furthermore, we show that the free boundary $\partial$ {u > 0} is locally Lipschitz continuous graph provided that $v = e_N$, the direction of $x_N$ coordinate axis and $\partial_{z}u\geq 0$. The latter monotonicity assumption in $z$ variable can be easily obtained for a suitable boundary condition.

2016, 36(1): 279-302
doi: 10.3934/dcds.2016.36.279

*+*[Abstract](456)*+*[PDF](470.8KB)**Abstract:**

The bang-bang property of time optimal controls for a semilinear parabolic equation, with homogeneous Dirichlet boundary condition and distributed controls acting on an open subset of the domain is established. This relies on an observability estimate from a measurable set in time for a linear parabolic equation, with potential depending on both space and time variables. The proof of the bang-bang property relies on a Kakutani fixed point argument.

2016, 36(1): 303-321
doi: 10.3934/dcds.2016.36.303

*+*[Abstract](292)*+*[PDF](460.1KB)**Abstract:**

Measure contraction properties are generalizations of the notion of Ricci curvature lower bounds in Riemannian geometry to more general metric measure spaces. In this paper, we give sufficient conditions for a Sasakian manifold equipped with a natural sub-Riemannian distance to satisfy these properties. Moreover, the sufficient conditions are defined by the Tanaka-Webster curvature. This generalizes the earlier work in [2] for the three dimensional case and in [19] for the Heisenberg group. To obtain our results we use the intrinsic Jacobi equations along sub-Riemannian extremals, coming from the theory of canonical moving frames for curves in Lagrangian Grassmannians [24,25]. The crucial new tool here is a certain decoupling of the corresponding matrix Riccati equation. It is also worth pointing out that our method leads to exact formulas for the measure contraction in the case of the corresponding homogeneous models in the considered class of sub-Riemannian structures.

2016, 36(1): 323-344
doi: 10.3934/dcds.2016.36.323

*+*[Abstract](381)*+*[PDF](621.0KB)**Abstract:**

An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $\beta$-shift (transformation) and an intermediate $\beta$-shift (transformation), for a fixed $\beta \in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{\beta,\alpha} \colon x \mapsto \beta x + \alpha \bmod 1$ for which the corresponding intermediate $\beta$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(\beta,\alpha)$ such that the intermediate $\beta$-shift associated with $T_{\beta, \alpha}$ is a subshift of finite type. It is also proved that these maps $T_{\beta,\alpha}$ are not transitive. This is in contrast to the situation for the corresponding greedy and lazy $\beta$-shifts and $\beta$-transformations, for which both of the two properties do not hold.

2016, 36(1): 345-360
doi: 10.3934/dcds.2016.36.345

*+*[Abstract](555)*+*[PDF](418.1KB)**Abstract:**

For almost periodic differential systems $\dot x= \varepsilon f(x,t,\varepsilon)$ with $x\in \mathbb{C}^n$, $t\in \mathbb{R}$ and $\varepsilon>0$ small enough, we get a polynomial normal form in a neighborhood of a hyperbolic singular point of the system $\dot x= \varepsilon \lim_{T \to \infty} \frac {1} {T} \int_0^T f(x,t,0) dt$, if its eigenvalues are in the Poincaré domain. The normal form linearizes if the real part of the eigenvalues are non--resonant.

2016, 36(1): 361-370
doi: 10.3934/dcds.2016.36.361

*+*[Abstract](358)*+*[PDF](341.1KB)**Abstract:**

We characterize the

*supercyclic*behavior of sequences of operators in a $C_0$-semigroup whose index set is a sector $\Delta$ in the complex plane $\mathbb{C}$.

2016, 36(1): 371-402
doi: 10.3934/dcds.2016.36.371

*+*[Abstract](384)*+*[PDF](606.3KB)**Abstract:**

In this paper, we consider the well-posedness of the Cauchy problem of the 3D incompressible nematic liquid crystal system with initial data in the critical Besov space $\dot{B}^{\frac{3}{p}-1}_{p,1}(\mathbb{R}^{3})\times \dot{B}^{\frac{3}{q}}_{q,1}(\mathbb{R}^{3})$ with $1< p<\infty$, $1\leq q<\infty$ and \begin{align*} -\min\{\frac{1}{3},\frac{1}{2p}\}\leq \frac{1}{q}-\frac{1}{p}\leq \frac{1}{3}. \end{align*} In particular, if we impose the restrictive condition $1< p<6$, we prove that there exist two positive constants $C_{0}$ and $c_{0}$ such that the nematic liquid crystal system has a unique global solution with initial data $(u_{0},d_{0}) = (u^{h}_{0}, u^{3}_{0}, d_{0})$ which satisfies \begin{align*} ((1+\frac{1}{\nu\mu})\|d_{0}-\overline{d}_{0}\|_{\dot{B}^{\frac{3}{q}}_{q,1}}+ \frac{1}{\nu}\|u_{0}^{h}\|_{\dot{B}^{\frac{3}{p}-1}_{p,1}}) \exp\left\{\frac{C_{0}}{\nu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{\frac{3}{p}-1}_{p,1}}+\frac{1}{\mu})^{2}\right\}\leq c_{0}, \end{align*} where $\overline{d}_{0}$ is a constant vector with $|\overline{d}_{0}|=1$. Here $\nu$ and $\mu$ are two positive viscosity constants.

2016, 36(1): 403-422
doi: 10.3934/dcds.2016.36.403

*+*[Abstract](305)*+*[PDF](445.8KB)**Abstract:**

For a given finite directed graph $G$, there are two types of Markov-Dyck shifts, the Markov-Dyck shift $D_G^V$ of vertex type and the Markov-Dyck shift $D_G^E$ of edge type. It is shown that, if $G$ does not have multi-edges, the former is a finite-to-one factor of the latter, and they have the same topological entropy. An expression for the zeta function of a Markov-Dyck shift of vertex type is given. It is different from that of the Markov-Dyck shift of edge type.

2016, 36(1): 423-450
doi: 10.3934/dcds.2016.36.423

*+*[Abstract](409)*+*[PDF](614.9KB)**Abstract:**

Without question, the dichotomy spectrum is a central tool in the stability, qualitative and geometric theory of nonautonomous dynamical systems. In this context, when dealing with time-variant linear equations having triangular coefficient matrices, their dichotomy spectrum associated to the whole time axis is not fully determined by the diagonal entries. This is surprising because such a behavior differs from both the half line situation, as well as the classical autonomous and periodic cases. At the same time triangular problems occur in various applications and particularly numerical techniques.

Based on operator-theoretical tools, this paper provides various sufficient and verifiable criteria to obtain a corresponding

*diagonal significance*for finite-dimensional difference equations in the following sense: Spectral and continuity properties of the diagonal elements extend to the whole triangular system.

2016, 36(1): 451-467
doi: 10.3934/dcds.2016.36.451

*+*[Abstract](306)*+*[PDF](807.9KB)**Abstract:**

Pyragas control is a widely used time-delayed feedback control for the stabilization of periodic orbits in dynamical systems. In this paper we investigate how we can use equivariance to eliminate restrictions of Pyragas control, both to select periodic orbits for stabilization by their spatio-temporal pattern and to render Pyragas control possible at all for those orbits. Another important aspect is the optimization of equivariant Pyragas control, i.e. to construct larger control regions. The ring of $n$ identical Stuart-Landau oscillators coupled diffusively in a bidirectional ring serves as our model.

2016, 36(1): 469-479
doi: 10.3934/dcds.2016.36.469

*+*[Abstract](382)*+*[PDF](383.3KB)**Abstract:**

Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $\mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, \mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $\Lambda$ is center topologically mixing then $f|_{\Lambda}$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $ h(f)\leq h(f,\mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $\mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.

2016, 36(1): 481-497
doi: 10.3934/dcds.2016.36.481

*+*[Abstract](373)*+*[PDF](478.3KB)**Abstract:**

This paper deals with the weak error estimates of the exponential Euler method for semi-linear stochastic partial differential equations (SPDEs). A weak error representation formula is first derived for the exponential integrator scheme in the context of truncated SPDEs. The obtained formula that enjoys the absence of the irregular term involved with the unbounded operator is then applied to a parabolic SPDE. Under certain mild assumptions on the nonlinearity, we treat a full discretization based on the spectral Galerkin spatial approximation and provide an easy weak error analysis, which does not rely on Malliavin calculus.

2016, 36(1): 499-508
doi: 10.3934/dcds.2016.36.499

*+*[Abstract](592)*+*[PDF](386.5KB)**Abstract:**

We study the existence of a radial sign-changing solution for the stationary fractional Schrödinger equation \begin{equation}\nonumber (-\Delta)^\alpha u + u=|u|^{p-1}u,\ x\in \mathbb{R}^N, N\geq 2.~~~~~~~~~~~~~{\rm (FS)} \end{equation} where $\alpha\in (0,1)$ and $p\in(1,2_\alpha^*-1)$, $2_\alpha^*=\frac{2N}{N-2\alpha}$. By using Brouwer degree theory and variational method, we prove that there exists a radial sign-changing solution of (FS).

2016, 36(1): 509-540
doi: 10.3934/dcds.2016.36.509

*+*[Abstract](569)*+*[PDF](409.6KB)**Abstract:**

In this paper, we first develop the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as behavior $0$ NMS flows). It mainly contains the following several directions:

1. we use WLG to list behavior $0$ NMS flows on $S^3$;

2. with the help of WLG, comparing with Wada's algorithm, we provide a direct description about the (indexed) link of behavior $0$ NMS flows;

3. to overcome the weakness that WLG can't decide topologically equivalent class, we give a simplified Umanskii Theorem to decide when two behavior $0$ NMS flows on $S^3$ are topological equivalence;

4. under these theories, we classify (up to topological equivalence) all behavior 0 NMS flows on $S^3$ with periodic orbits number no more than 4.

2016, 36(1): 541-570
doi: 10.3934/dcds.2016.36.541

*+*[Abstract](279)*+*[PDF](487.8KB)**Abstract:**

We compare the spectra of

*Dynnikov matrices*with the spectra of the

*train track transition matrices*of a given pseudo-Anosov braid on the finitely punctured disk, and show that these matrices are isospectral up to roots of unity and zeros under some particular conditions. It is shown, via examples, that Dynnikov matrices are much easier to compute than transition matrices, and so yield data that was previously inaccessible.

2017 Impact Factor: 1.179

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