# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

February 2015 , Volume 35 , Issue 2

Special issue on advances and applications in qualitative studies of dynamics

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2015, 35(2): i-ii doi: 10.3934/dcds.2015.35.2i +[Abstract](1248) +[PDF](104.8KB)
Abstract:
The idea of this special issue was to gather a number of articles on various mathematical tools in studies on qualitative aspects of dynamics. Of course not every important topic could be included in this issue due to space limitations. First of all we decided to focus on qualitative properties of topological dynamics by various tools coming from different fields such as functional and real analysis, measure-theory and topology itself. We aimed to present various aspects of such analysis, and when possible, present concrete applications of developed general (theoretical in nature) methodology. This should additionally highlight close connections between pure and applied mathematics.

2015, 35(2): 595-615 doi: 10.3934/dcds.2015.35.595 +[Abstract](1197) +[PDF](466.7KB)
Abstract:
We study the analyticity of the semigroups generated by some classes of degenerate second order differential operators in the space of continuous function on a domain with corners. These semigroups arise from the theory of dynamics of populations.
2015, 35(2): 617-635 doi: 10.3934/dcds.2015.35.617 +[Abstract](1537) +[PDF](689.7KB)
Abstract:
Multiple time scales are common in population models with age and space structure, where they are a reflection of often different rates of demographic and migratory processes. This makes the models singularly perturbed and allows for their aggregation which, while significantly reducing their complexity, does not alter their essential dynamic properties. There are several methods of aggregation of such models. In this paper we shall show how the Trotter-Kato-Sova-Kurtz theory developed to analyze convergence of $C_0$-semigroups can be used in this field. The paper also extends some of the previous results by considering reducible migration matrices which are important in modelling populations living in geographically patched areas with restricted communication between the patches.
2015, 35(2): 637-652 doi: 10.3934/dcds.2015.35.637 +[Abstract](1203) +[PDF](389.0KB)
Abstract:
We study the integration operator, the differentiation operator and more general differential operators on radial Fréchet or (LB) Hörmander algebras of entire functions. We analyze when these operators are power bounded, hypercyclic and (uniformly) mean ergodic.
2015, 35(2): 653-668 doi: 10.3934/dcds.2015.35.653 +[Abstract](1274) +[PDF](438.8KB)
Abstract:
The Hyperbolic Heat Transfer Equation describes heat processes in which extremely short periods of time or extreme temperature gradients are involved. It is already known that there are solutions of this equation which exhibit a chaotic behaviour, in the sense of Devaney, on certain spaces of analytic functions with certain growth control. We show that this chaotic behaviour still appears when we add a source term to this equation, i.e. in the Hyperbolic Bioheat Equation. These results can also be applied for the Wave Equation and for a higher order version of the Hyperbolic Bioheat Equation.
2015, 35(2): 669-701 doi: 10.3934/dcds.2015.35.669 +[Abstract](1418) +[PDF](1202.4KB)
Abstract:
We study the number of limit cycles and the bifurcation diagram in the Poincaré sphere of a one-parameter family of planar differential equations of degree five $\dot {\bf x}=X_b({\bf x})$ which has been already considered in previous papers. We prove that there is a value $b^*>0$ such that the limit cycle exists only when $b\in(0,b^*)$ and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length $27/1000$ where $b^*$ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic curves without contact for the flow of the differential equation. These curves are obtained using analytic information about the separatrices of the infinite critical points of the vector field. To prove that the Bendixson--Dulac Theorem works we develop a method for studying whether one-parameter families of polynomials in two variables do not vanish based on the computation of the so called double discriminant.
2015, 35(2): 703-723 doi: 10.3934/dcds.2015.35.703 +[Abstract](1257) +[PDF](437.9KB)
Abstract:
Let $X,U$ and $Z$ be Banach spaces such that $Z\subset X$ (with continuous and dense embedding), $L:Z\to X$ be a closed linear operator and consider closed linear operators $G,M:Z\to U$. Putting conditions on $G$ and $M$ we show that the operator $\mathcal{A}=L$ with domain $D(\mathcal{A})=\big\{z\in Z:Gz=Mz\big\}$ generates a $C_0$-semigroup on $X$. Moreover, we give a variation of constants formula for the solution of the following inhomogeneous problem \begin{align*} \begin{cases} \dot{z}(t)=L z(t)+f(t),& t\ge 0,\cr G z(t)=Mz(t)+g(t),& t\ge 0,\cr z(0)=z^0. \end{cases} \end{align*} Several examples will be given, in particular a heat equation with distributed unbounded delay at the boundary condition.
2015, 35(2): 725-740 doi: 10.3934/dcds.2015.35.725 +[Abstract](1383) +[PDF](412.8KB)
Abstract:
This paper is devoted to studying the localization of mixing property via Furstenberg families. It is shown that there exists some $\mathcal{F}_{pubd}$-mixing set in every dynamical system with positive entropy, and some $\mathcal{F}_{ps}$-mixing set in every non-PI minimal system.
2015, 35(2): 741-755 doi: 10.3934/dcds.2015.35.741 +[Abstract](1458) +[PDF](411.4KB)
Abstract:
Let $f:I\rightarrow I$ be a continuous map of a compact interval $I$ and $C(I)$ be the space of all compact subintervals of $I$ with the Hausdorff metric. We investigate chain transitivity of induced maps on subcontinua of $C(I)$. In particular, we prove the following theorem: Let $\mathcal{M}$ be a subcontinuum of $C(I)$ having at most countably many partitioning points. Then, the induced map $\mathcal{F}:C(I)\to C(I)$ $($i.e. $\mathcal{F}(A):=\{f(x):x\in A\}$ for each $A \in C(I)$$)$ is chain transitive on $\mathcal{M}$ iff $\mathcal{F}^{2}\vert_{\mathcal{M}}=Id$.
2015, 35(2): 757-770 doi: 10.3934/dcds.2015.35.757 +[Abstract](1640) +[PDF](417.0KB)
Abstract:
This paper is devoted to the ergodic-theoretical approach to chaos, which is based on the existence of invariant mixing measures supported on the whole space. As an example of application of the general theory we prove that there exists an invariant mixing measure with respect to the differentiation operator on the space of entire functions. From this theorem it follows the existence of universal entire functions and other chaotic properties of this transformation.
2015, 35(2): 771-792 doi: 10.3934/dcds.2015.35.771 +[Abstract](1495) +[PDF](464.9KB)
Abstract:
By a result of Blokh from 1984, every transitive map of a tree has the relative specification property, and so it has finite decomposition ideal, positive entropy and dense periodic points. In this paper we construct a transitive dendrite map with infinite decomposition ideal and a unique periodic point. Basically, the constructed map is (with respect to any non-atomic invariant measure) a measure-theoretic extension of the dyadic adding machine. Together with an example of Hoehn and Mouron from 2013, this shows that transitivity on dendrites is much more varied than that on trees.

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