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<p>In this paper, we study the existence of analytic invariant curves for McMillan maps with mixed nonlinearity. Analytic invariant curves of this map are discussed by locally reducing its invariance functional equation to another the so-called auxiliary <i>q</i>-difference equation and by constructing solutions in uniformly convergent power series for the latter equation. For the auxiliary <i>q</i>-difference equation, using classical majorant series method we obtain the following results: (1) We prove the existence of local invertible analytic solutions when <i>q</i> lies not on the unit circle (see Proposition&#xa0;<InternalRef RefID="FPar5">2.1</InternalRef>). (2) We prove the existence of local invertible analytic solutions when <i>q</i> lies on the unit circle but is not a root of unity under the Brjuno condition (see Proposition&#xa0;<InternalRef RefID="FPar15">2.7</InternalRef>). (3) We prove the existence of local invertible analytic solutions when <i>q</i> lies on the unit circle and is a root of unity, which is called the resonance case (see Proposition&#xa0;<InternalRef RefID="FPar18">2.9</InternalRef>). Main results for analytic invariant curves of the map considered in this paper are direct corollaries of Propositions&#xa0;<InternalRef RefID="FPar5">2.1</InternalRef>, <InternalRef RefID="FPar15">2.7</InternalRef>,&#xa0;<InternalRef RefID="FPar18">2.9</InternalRef>, respectively.</p>

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Construction of Analytic Invariant Curves for McMillan Maps with Mixed Nonlinearity

  • 期刊论文

  • 发表日期：

    2026年5月15日

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  • Jianguo Si,
  • Linli Liu

摘要

In this paper, we study the existence of analytic invariant curves for McMillan maps with mixed nonlinearity. Analytic invariant curves of this map are discussed by locally reducing its invariance functional equation to another the so-called auxiliary q-difference equation and by constructing solutions in uniformly convergent power series for the latter equation. For the auxiliary q-difference equation, using classical majorant series method we obtain the following results: (1) We prove the existence of local invertible analytic solutions when q lies not on the unit circle (see Proposition 2.1). (2) We prove the existence of local invertible analytic solutions when q lies on the unit circle but is not a root of unity under the Brjuno condition (see Proposition 2.7). (3) We prove the existence of local invertible analytic solutions when q lies on the unit circle and is a root of unity, which is called the resonance case (see Proposition 2.9). Main results for analytic invariant curves of the map considered in this paper are direct corollaries of Propositions 2.1, 2.7, 2.9, respectively.