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Abstract <p> The paper studies the spectra (sets of values) of wandering indicators of solutions ofdifferential systems. We construct two-dimensional systems with a nonlinearity of a givenarbitrarily high order of smallness in a neighborhood of the origin such that<OrderedList> <ListItem> <ItemNumber>(i)</ItemNumber> <ItemContent> <p>All solutions are infinitely extendable to the right. </p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(ii)</ItemNumber> <ItemContent> <p>Any of their spectra of wandering indicators can coincide with either the interval<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10625_2025_2780_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\([0,1]\)</EquationSource> </InlineEquation> or any prescribed nonempty set of rationalnumbers on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10625_2025_2780_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\([0,1]\)</EquationSource> </InlineEquation>. </p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(iii)</ItemNumber> <ItemContent> <p>For the linear first approximation systems, the corresponding spectra are singletons.</p> </ItemContent> </ListItem> </OrderedList></p> <p>Moreover, these nonlinear two-dimensional systems have the same spectra ofwandering indicators as their restrictions to the direct product of any open neighborhood of zeroon the phase plane by the time half-line.</p>

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On the Difference in Cardinalities of the Spectra of Exact and Absolute Wandering Indicators Between a Two-Dimensional Nonlinear System and the First Approximation System

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • A. Kh. Stash,
  • N. A. Loboda

摘要

Abstract

The paper studies the spectra (sets of values) of wandering indicators of solutions ofdifferential systems. We construct two-dimensional systems with a nonlinearity of a givenarbitrarily high order of smallness in a neighborhood of the origin such that (i)

All solutions are infinitely extendable to the right.

(ii)

Any of their spectra of wandering indicators can coincide with either the interval \([0,1]\) or any prescribed nonempty set of rationalnumbers on \([0,1]\) .

(iii)

For the linear first approximation systems, the corresponding spectra are singletons.

Moreover, these nonlinear two-dimensional systems have the same spectra ofwandering indicators as their restrictions to the direct product of any open neighborhood of zeroon the phase plane by the time half-line.