Exploring Limit Cycles of Differential Equations through Information Geometry Unveils the Solution to Hilbert’s 16th Problem

The detection of limit cycles of differential equations poses a challenge due to the type of the nonlinear system, the regime of interest, and the broader context of applicable models. Consequently, attempts to solve Hilbert’s sixteenth problem on the maximum number of limit cycles of polynomial differential equations have been uniformly unsuccessful due to failing results and their lack of consistency. Here, the answer to this problem is finally obtained through information geometry, in which the Riemannian metrical structure of the parameter space of differential equations is investigated with the aid of the Fisher information metric and its scalar curvature R. We find that the total number of divergences of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi mathvariant="normal">R</mi><mo>|</mo></mrow></semantics></math></inline-formula> to infinity provides the maximum number of limit cycles of differential equations. Additionally, we demonstrate that real polynomial systems of degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> have the maximum number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mn>4</mn><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>−</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> limit cycles. The research findings highlight the effectiveness of geometric methods in analyzing complex systems and offer valuable insights across information theory, applied mathematics, and nonlinear dynamics. These insights may pave the way for advancements in differential equations, presenting exciting opportunities for future developments.