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Register a new userOn existence and uniqueness of the carrying simplex for competitive dynamical systems
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Authors
Morris W. Hirsch
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Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.
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We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of 33 stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes 1-25 and 33; there is always a heteroclinic cycle in class 27; Neimark-Sacker bifurcations may occur in classes 26-31 but cannot occur in class 32. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes 29,31,33 and those in class 27 with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.
This paper is devoted to the quantitative study of the attractive velocity of generalized attractors for infinite-dimensional dynamical systems. We introduce the notion of~$varphi$-attractor whose attractive speed is characterized by a general non-negative decay function~$varphi$, and prove that~$varphi$-decay with respect to noncompactness measure is a sufficient condition for a dissipitive system to have a~$varphi$-attractor. Furthermore, several criteria for~$varphi$-decay with respect to noncompactness measure are provided. Finally, as an application, we establish the existence of a generalized exponential attractor and the specific estimate of its attractive velocity for a semilinear wave equation with a critical nonlinearity.
We investigate Takagi-type functions with roughness parameter $gamma$ that are Holder continuous with coefficient $H=frac{loggamma}{log eh}.$ Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We show that the SBR measure is absolutely continuous for large enough $gamma$. Dually, where duality is related to time reversal, we prove that for large enough $gamma$ a version of the Takagi-type curve centered around fibers of the associated stable manifold possesses a square integrable local time.
Let $(X,mathcal{B},mu)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: begin{eqnarray*} f = g - g circ T end{eqnarray*} where $f in L^p$ and $T$ is ergodic invertible measure preserving on $(X, mathcal{B}, mu )$. We extend previous results by showing for any measurable $f$ that is non-zero on a set of positive measure, the class of measure preserving $T$ with a measurable solution $g$ is meager (including the case where $int_X f dmu = 0$). From this fact, a natural question arises: given $f$, does there always exist a solution pair $T$ and $g$? In regards to this question, our main results are: (i) Given measurable $f$, there exists an ergodic invertible measure preserving transformation $T$ and measurable function $g$ such that $f(x) = g(x) - g(Tx)$ for a.e. $xin X$, if and only if $int_{f > 0} f dmu = - int_{f < 0} f dmu$ (whether finite or $infty$). (ii) Given mean-zero $f in L^p$ for $p geq 1$, there exists an ergodic invertible measure preserving $T$ and $g in L^{p-1}$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x in X$. (iii) In some sense, the previous existence result is the best possible. For $p geq 1$, there exist mean-zero $f in L^p$ such that for any ergodic invertible measure preserving $T$ and any measurable $g$ such that $f(x) = g(x) - g(Tx)$ a.e., then $g otin L^q$ for $q > p - 1$. Also, we show this situation is generic for mean-zero $f in L^p$. Finally, it is shown that we cannot expect finite moments for solutions $g$, when $f in L^1$. In particular, given any $phi : mathbb{R} to mathbb{R}$ such that $lim_{xto infty} phi (x) = infty$, there exist mean-zero $f in L^1$ such that for any solutions $T$ and $g$, the transfer function $g$ satisfies: begin{eqnarray*} int_{X} phi big( | g(x) | big) dmu = infty. end{eqnarray*}
We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, ErdH{o}s, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Levy on the regular continued fraction expansion of irrational numbers in $(0,1)$.
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