Fractional scalar products as probes of chaos and fractal dimension

Abstract We develop a fractional-weighted functional-analytic framework for the analysis of chaotic dynamics in which the governing equations remain classical while the geometry of the underlying Hilbert space is modified. Specifically, we introduce a family of fractional scalar products with singular weights derived from the Riemann–Liouville kernel, generating weighted Hilbert spaces that emphasize late-time dynamics and long-term correlations. Within this framework, the fractional parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> plays a dual role by controlling temporal localization in the scalar product and acting as an effective probe of dynamical complexity. By embedding trajectories of the classical Lorenz system into these spaces, we show that the value $$\alpha _{\min }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mo>min</mml:mo> </mml:msub> </mml:math> minimizing the normalized fractional norm exhibits a clear nonlinear correlation with the Kaplan–Yorke dimension $$D_{KY}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>KY</mml:mi> </mml:mrow> </mml:msub> </mml:math> of the attractor, thereby establishing $$\alpha _{\min }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mo>min</mml:mo> </mml:msub> </mml:math> as a functional proxy for fractal complexity without modifying the underlying dynamics. To support analysis and computation, we construct orthogonal and complete basis systems adapted to the fractional geometry, including weighted Gram–Schmidt bases and Jacobi polynomial expansions, which enable efficient spectral approximation of chaotic signals and reveal intrinsic temporal asymmetries not captured by standard $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> representations. The proposed approach provides new analytical and spectral tools for detecting bifurcations, quantifying chaotic complexity, and representing fractal structure, offering a complementary alternative to existing methods based on fractional-order dynamical models.