Analytical Characterization of Epileptic Dynamics in a Bistable System

The paper introduces the planar system ẋ = −ωy + x(σ + 2ab(x² + y²) − b(x² + y²)²), ẏ = ωx + y(σ + 2ab(x² + y²) − b(x² + y²)²), derives its polar form ṙ = r(σ + 2ab r² − b r⁴), θ̇ = ω, and states that in the bistable regime −a²b < σ < 0, the basins are Ae = {x² + y² < a − γ0} and Aℓ = {x² + y² > a − γ0}, with C2 as the separatrix and C1 stable; this is Theorem 1 with a proof via Lyapunov/LaSalle arguments (and Appendix details) . The candidate solution independently reduces to polar coordinates, analyzes the scalar r-dynamics, identifies the same equilibria r = 0, r = √(a ± γ0), proves their stability via 1D phase-line analysis, shows positive invariance of the regions separated by r = √(a − γ0), and establishes convergence to e0 or C1 accordingly; it also notes the period 2π/ω, consistent with the paper’s frequency statement . Aside from a minor notational shortcut when writing F′(r) = 4br²(a − r²) (valid only at nonzero equilibria where σ + 2abr² − br⁴ = 0), the model’s argument is correct and complete. Hence, both are correct, using different proof styles (Lyapunov/LaSalle vs. scalar radial dynamics).