Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka...Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka-Volterra dynamical systems exhibit all of the dynamics of the lower dimensional systems and a great deal more. In fact and in particular, there are dynamical features including analogs of flip bifurcations, Neimark-Sacker bifurcations and chaotic strange attracting sets that are essentially three-dimensional. Among these are new generalizations of Neimark-Sacker bifurcations and novel chaotic strange attractors with distinctive candy cane type shapes. Several of these dynamical are investigated in detail using both analytical and simulation techniques.展开更多
We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the...We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the invariance entropy for uncertain control systems.Moreover,we present four versions of measure-theoretic invariance entropies,give an inverse variational principle of the measure-theoretic Bowen invariance entropy,and obtain four positive variational principles of invariance entropies of subsets with finite elements for uncertain control systems.展开更多
文摘Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka-Volterra dynamical systems exhibit all of the dynamics of the lower dimensional systems and a great deal more. In fact and in particular, there are dynamical features including analogs of flip bifurcations, Neimark-Sacker bifurcations and chaotic strange attracting sets that are essentially three-dimensional. Among these are new generalizations of Neimark-Sacker bifurcations and novel chaotic strange attractors with distinctive candy cane type shapes. Several of these dynamical are investigated in detail using both analytical and simulation techniques.
基金supported by National Natural Science Foundation of China(Grant Nos.12171492 and 12201135).
文摘We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the invariance entropy for uncertain control systems.Moreover,we present four versions of measure-theoretic invariance entropies,give an inverse variational principle of the measure-theoretic Bowen invariance entropy,and obtain four positive variational principles of invariance entropies of subsets with finite elements for uncertain control systems.
