KR-Regulated Nonlinear Parabolic and Fractional Evolution Equations: Attractor Scaling Laws, Critical Thresholds, and Computational Efficiency
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2026
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| _version_ | 1866902005365080064 |
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| author | Pasupuleti, Ramakrishna |
| author_facet | Pasupuleti, Ramakrishna |
| contents | <p>We introduce and analyse a dual-parameter excitation–regulation (KR) framework for the nonlinear parabolic equation u_t = Δu + Ku − Ru³ with E(u) = u, G(u) = u³ (logistic-type operators) on bounded domains with Dirichlet boundary conditions. A linearisation argument proves that the equilibrium u = 0 loses stability precisely when K exceeds the first Dirichlet eigenvalue K_c = λ₁(Ω): in 1D (0,1) this gives K_c = π² ≈ 9.87; in 2D (0,1)² it gives K_c = 2π² ≈ 19.74. For K > K_c the system maintains a nontrivial global attractor with explicit scaling bound R* ≤ C√((K − λ₁)/R). This connects directly to Allen–Cahn dynamics — Allen–Cahn (K = 0) always decays to zero, while the KR system sustains regulated bounded steady states. The framework extends to Caputo fractional-order dynamics (α ∈ (0,1]) via an IMEX-L1 scheme, proving that fractional memory alters convergence speed (T_c grows as α → 0) but leaves R* invariant to machine precision. Seven computational studies verify: (i) EOC = 2.0000 via manufactured solution; (ii) scaling law slope 0.512 ± 0.003 across 120 parameter pairs; (iii) sharp K_c transition in both 1D and 2D; (iv) R* = 1.86863 identical for five IC types (smooth, small, random, discontinuous, multi-mode); (v) fractional α invariance to < 10⁻¹²; (vi) IMEX speedup 5,212× at N = 400; (vii) applications to population dynamics, chemical reaction-diffusion, and excitation–inhibition balance in neuroscience.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19050953 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | KR-Regulated Nonlinear Parabolic and Fractional Evolution Equations: Attractor Scaling Laws, Critical Thresholds, and Computational Efficiency Pasupuleti, Ramakrishna <p>We introduce and analyse a dual-parameter excitation–regulation (KR) framework for the nonlinear parabolic equation u_t = Δu + Ku − Ru³ with E(u) = u, G(u) = u³ (logistic-type operators) on bounded domains with Dirichlet boundary conditions. A linearisation argument proves that the equilibrium u = 0 loses stability precisely when K exceeds the first Dirichlet eigenvalue K_c = λ₁(Ω): in 1D (0,1) this gives K_c = π² ≈ 9.87; in 2D (0,1)² it gives K_c = 2π² ≈ 19.74. For K > K_c the system maintains a nontrivial global attractor with explicit scaling bound R* ≤ C√((K − λ₁)/R). This connects directly to Allen–Cahn dynamics — Allen–Cahn (K = 0) always decays to zero, while the KR system sustains regulated bounded steady states. The framework extends to Caputo fractional-order dynamics (α ∈ (0,1]) via an IMEX-L1 scheme, proving that fractional memory alters convergence speed (T_c grows as α → 0) but leaves R* invariant to machine precision. Seven computational studies verify: (i) EOC = 2.0000 via manufactured solution; (ii) scaling law slope 0.512 ± 0.003 across 120 parameter pairs; (iii) sharp K_c transition in both 1D and 2D; (iv) R* = 1.86863 identical for five IC types (smooth, small, random, discontinuous, multi-mode); (v) fractional α invariance to < 10⁻¹²; (vi) IMEX speedup 5,212× at N = 400; (vii) applications to population dynamics, chemical reaction-diffusion, and excitation–inhibition balance in neuroscience.</p> |
| title | KR-Regulated Nonlinear Parabolic and Fractional Evolution Equations: Attractor Scaling Laws, Critical Thresholds, and Computational Efficiency |
| url | https://doi.org/10.5281/zenodo.19050953 |