Fractal-Type Scaling Laws and Multiscale Oscillatory Dynamics in a Four-Dimensional Faraday Disk Dynamo System: Zero-Hopf Bifurcation Analysis via Averaging Theory

This paper presents a rigorous analytical study of fractal-type scaling laws and multiscale oscillatory dynamics arising from a zero-Hopf bifurcation in a four-dimensional Faraday disk dynamo system originally introduced by Hide and Moroz. The system can be written in the standard form [Formula: see text] where [Formula: see text] is the linearization matrix at the equilibrium and [Formula: see text] contains the nonlinear terms. Zero-Hopf bifurcations occur when [Formula: see text] has a simple zero eigenvalue and a pair of purely imaginary eigenvalues [Formula: see text], leading to slow–fast interactions that generate complex oscillatory regimes, scale-dependent transitions, and self-similar structures in the associated phase space, characteristic of fractal-type behavior. By using first-order averaging theory, the dynamics near the zero-Hopf point can be reduced to the averaged system: [Formula: see text] where [Formula: see text] represents the amplitude of oscillations in the center manifold and [Formula: see text] represents the slow mode associated with the zero eigenvalue. Analytical conditions for the bifurcation of small-amplitude periodic orbits can be expressed as [Formula: see text] with solutions [Formula: see text] corresponding to families of periodic solutions whose stability is determined by the Jacobian of the averaged system: [Formula: see text] From a broader perspective, the derived results demonstrate how nonlinear electromechanical dynamo models exhibit multiscale, self-organized oscillatory behavior governed by intrinsic scaling laws. The emergence of parameter-dependent oscillatory amplitudes and slow–fast phase interactions highlights the presence of fractal-type structures in the phase space near critical bifurcation regimes. The analytical framework developed here provides a solid mathematical foundation for previously observed numerical phenomena and offers a basis for future studies aimed at identifying and characterizing complex multiscale and fractal dynamical patterns in high-dimensional nonlinear systems.