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<p>In this paper, we present a generalization of an algebraic criterion previously established in Messias, Silva, Internat. J. Bifur. Chaos Appl. Sci. Engrg. <b>28</b>, 1830006 (2018), aimed at establishing the non-chaotic behavior of three-dimensional polynomial differential systems. By employing tools from the Darboux Theory of Integrability, we develop sufficient conditions to guarantee that a given system does not exhibit chaotic dynamics. Our main result ensures that if a system possesses a finite number of singularities and a set of invariant algebraic surfaces whose cofactors satisfy a specific linear relation, then the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-limit sets of all orbits are contained within the union of these surfaces or tend toward infinity, thereby ruling out chaotic behavior. Several new classes of non-chaotic polynomial systems are constructed, including systems with invariant quadrics such as cylinders, paraboloids, and cones. These results extend the authors’ previous findings, expanding the range of differential systems for which non-chaotic behavior can be analytically demonstrated, as an alternative to commonly used numerical methods for chaos detection. Our approach builds upon foundational work on invariant algebraic surfaces and is illustrated by explicit systems that satisfy the proposed criteria, reinforcing the effectiveness of algebraic invariants in the qualitative analysis of dynamical systems.</p>

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An Algebraic Criterion for the Determination of Non-chaotic Behavior in Three-Dimensional Polynomial Differential Systems

  • 期刊论文

  • 发表日期：

    2026年5月15日

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  • Rafael Paulino Silva,
  • Marcelo Messias

摘要

In this paper, we present a generalization of an algebraic criterion previously established in Messias, Silva, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28, 1830006 (2018), aimed at establishing the non-chaotic behavior of three-dimensional polynomial differential systems. By employing tools from the Darboux Theory of Integrability, we develop sufficient conditions to guarantee that a given system does not exhibit chaotic dynamics. Our main result ensures that if a system possesses a finite number of singularities and a set of invariant algebraic surfaces whose cofactors satisfy a specific linear relation, then the \(\alpha \) α - and \(\omega \) ω -limit sets of all orbits are contained within the union of these surfaces or tend toward infinity, thereby ruling out chaotic behavior. Several new classes of non-chaotic polynomial systems are constructed, including systems with invariant quadrics such as cylinders, paraboloids, and cones. These results extend the authors’ previous findings, expanding the range of differential systems for which non-chaotic behavior can be analytically demonstrated, as an alternative to commonly used numerical methods for chaos detection. Our approach builds upon foundational work on invariant algebraic surfaces and is illustrated by explicit systems that satisfy the proposed criteria, reinforcing the effectiveness of algebraic invariants in the qualitative analysis of dynamical systems.