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<p>Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our goal has been to define a meaningful concept of graph of any dynamical system. As a result, we base our definition of “chain graph” on “<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11466_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-chains”, defining both nodes and edges of the graph in terms of chains. In particular, nodes are often maximal limit sets and there is an edge between two nodes if there is a trajectory whose forward limit set is in one node and its backward limit set is in the other. Our initial goal was to prove that every “chain graph” of a dynamical system is, in some sense, connected, and we prove connectedness under mild hypotheses.</p>

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What is the graph of a dynamical system?

  • 期刊论文

  • 发表日期：

    2025年7月2日

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  • Chirag Adwani,
  • Roberto De Leo,
  • James Yorke

摘要

Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our goal has been to define a meaningful concept of graph of any dynamical system. As a result, we base our definition of “chain graph” on “ \(\varepsilon \) ε -chains”, defining both nodes and edges of the graph in terms of chains. In particular, nodes are often maximal limit sets and there is an edge between two nodes if there is a trajectory whose forward limit set is in one node and its backward limit set is in the other. Our initial goal was to prove that every “chain graph” of a dynamical system is, in some sense, connected, and we prove connectedness under mild hypotheses.