Strange Attractors in Fractional Differential Equations: A Topological Approach to Chaos and Stability

The paper’s core claim (Theorem 2) asserts an unconditional “spectral criterion for chaotic attractors” for Caputo systems: if eigenvalues at an equilibrium satisfy Re(λi) > πα/2 and the eigenspace yields a partial hyperbolic splitting, then the system has a strange attractor with nontrivial homoclinic orbits and positive topological entropy. This is stated and informally “proved” in the PDF, including reliance on a multiplicative ergodic theorem and a Ruelle–Pesin formula for fractional systems, and an appeal to Smale–Birkhoff in the fractional setting . However, multiple elements are incorrect or unsupported: (i) the stability/instability wedge for Caputo linear systems depends on the argument, not a real-part threshold (the paper repeatedly uses Re(λ) < 0 for stability and Re(λ) > 0 for instability) ; (ii) the “fractional Conley index” is misdefined as an alternating sum of homology dimensions (an Euler characteristic), and then used as a chaos detector ; (iii) the text asserts a fractional MET/Ruelle–Pesin and directly applies Smale–Birkhoff without constructing a valid Poincaré map in an appropriate (history) phase space or verifying the needed hypotheses . By contrast, the candidate solution correctly identifies the known spectral wedge |arg λ| ≷ απ/2 for Caputo systems, explains why local spectrum alone is insufficient to force chaotic attractors, and describes the additional global hypotheses (dissipativity and transverse homoclinic intersections, typically in an extended phase space) under which horseshoes and positive entropy follow. Hence the paper’s theorem is wrong/inadequately justified, whereas the model’s position reflects the established theory.