<p>The analytic normal forms for the nonlinear system <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\dot{x}=A(t)x+f(x,t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with linear part admitting mean exponential dichotomy are illustrated. As the preparatory step towards stability assessment, bifurcation analysis and chaos detection for weak hyperbolic systems, the major obstacles lie in achieving analyticity for normalization while accommodating the coexistence of hyperbolic and non-hyperbolic behaviors owing to error function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon (t,s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. An unconventional spectral concept is introduced to effectively promote regularity of normalization. By the invariance of mean dichotomy spectrum and homotopy method, the analytic equivalent jet class is provided within generalized Poincaré domain <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(a_1-\rho )(b_m+\rho )>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>-</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mi>m</mi> </msub> <mo>+</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we establish the Poincaré–Dulac and Poincaré type analytic normal forms via strongly mean hyperbolicity, enabling the maximal elimination of non-resonant terms in weak hyperbolic cases.</p>

Analytic normalization for weak hyperbolic systems with mean dichotomy spectrum

期刊论文
发表日期：

2026年2月18日

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Jiahui Feng,
Yong Li

摘要

The analytic normal forms for the nonlinear system $\dot{x}=A(t)x+f(x,t)$ $\dot{x} = A (t) x + f (x, t)$ with linear part admitting mean exponential dichotomy are illustrated. As the preparatory step towards stability assessment, bifurcation analysis and chaos detection for weak hyperbolic systems, the major obstacles lie in achieving analyticity for normalization while accommodating the coexistence of hyperbolic and non-hyperbolic behaviors owing to error function $\varepsilon (t,s)$ $ε (t, s)$ . An unconventional spectral concept is introduced to effectively promote regularity of normalization. By the invariance of mean dichotomy spectrum and homotopy method, the analytic equivalent jet class is provided within generalized Poincaré domain $(a_1-\rho )(b_m+\rho )>0$ $(a_{1} - ρ) (b_{m} + ρ) > 0$ . Furthermore, we establish the Poincaré–Dulac and Poincaré type analytic normal forms via strongly mean hyperbolicity, enabling the maximal elimination of non-resonant terms in weak hyperbolic cases.