Stable stationary solutions in reaction-diffusion systems consisting of a 1-D array of bistable cells LEONID P. NIZHNIK, IRINA L. NIZHNIK, MARTIN HASLER International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2002 In this paper we present the construction of stable stationary solutions in reaction–diffusion systems consisting of a 1-D array of bistable cells with a cubic nonlinearity and with a cubic-like piecewise-linear nonlinearity. Some periodic solutions, kinks, solitons are considered. While it is known that spatial chaos arises in such systems with small coupling constants, we show the existence of spatial chaos for an arbitrary value of the cell coupling constant, in the case of the piecewise-linear nonlinearity. The value of the spatial entropy is found. We also show the existence of stable spatially periodic (pattern) solutions that persist for large coupling constants.
Spectral Analysis of Self-Adjoint Momentum Operators with Nonlocal Potentials K Dębowska, I Nizhnik Ukrainian Mathematical Journal, 1-33 , 2025 2025.0 Citations: 1
Deterministic diffusion in dynamical systems with a tiled phase space I Nizhnik arXiv preprint arXiv:2501.00182 , 2024 2024.0
Spectral analysis of self-adjoint first order differential operators with nonlocal potentials K Dębowska, IL Nizhnik arXiv preprint arXiv:2312.15775 , 2023 2023.0 Citations: 1
Differential equations with bistable nonlinearity AM Samoilenko, IL Nizhnik Ukrainian Mathematical Journal 67 (4), 584-624 , 2015 2015.0 Citations: 12
Deterministic Diffusion L Nizhnik, I Nizhnik arXiv preprint arXiv:1501.00674 , 2015 2015.0 Citations: 1
Kinklike solutions of fourth-order differential equations with a cubic bistable nonlinearity AM Samoilenko, IL Nizhnik Differential Equations 50 (2), 203-212 , 2014 2014.0 Citations: 1
Sergio Albeverio S Albeverio, GA Galperin, IL Nizhnik, LP Nizhnik 2014.0
Periodic solutions of second-order differential equations with discontinuous nonlinearity I Nyzhnyk, Krasneeva Journal of Mathematical Sciences 191 (3) , 2013 2013.0 Citations: 14
Stable stationary solutions for a reaction–diffusion equation with a multi-stable nonlinearity I Nizhnik Physics Letters A 357 (4-5), 319-322 , 2006 2006.0 Citations: 9
Generalized billiards inside an infinite strip with periodic laws of reflection along the strip's boundaries SA Albeverio, GA Gal'perin, IL Nizhnik, LP Nizhnik Regular and Chaotic Dynamics 10 (3), 285-306 , 2005 2005.0 Citations: 4
Stable stationary solutions in reaction–diffusion systems consisting of a 1-d array of bistable cells LP Nizhnik, IL Nizhnik, M Hasler International Journal of Bifurcation and Chaos 12 (02), 261-279 , 2002 2002.0 Citations: 32
Spatial chaos in a fourth-order nonlinear parabolic equation S Albeverio, IL Nizhnik Physics Letters A 288 (5-6), 299-304 , 2001 2001.0 Citations: 28
Stable solutions of a system of two connected Chua cells IL Nizhnik Ukrainian Mathematical Journal 50 (11), 1789-1791 , 1998 1998.0
Stable difference scheme for a nonlinear Klein-Gordon equation IL Nizhnik Ukrainian Mathematical Journal 49 (6), 960-962 , 1997 1997.0 Citations: 1
Soliton attraction law IL Nizhnik Ukrainian Mathematical Journal 47 (5), 789-792 , 1995 1995.0
Publication: Regular and Chaotic Dynamics Pub Date: 2005 S Albeverio, G Galperin, IL Nizhnik, LP Nizhnik
Traveling avalanche waves in spatially discrete bistable reaction–diffusion systems L Nizhnik, M Hasler, I Nizhnik
MOST CITED SCHOLAR PUBLICATIONS
Stable stationary solutions in reaction–diffusion systems consisting of a 1-d array of bistable cells LP Nizhnik, IL Nizhnik, M Hasler International Journal of Bifurcation and Chaos 12 (02), 261-279 , 2002 2002.0 Citations: 32
Spatial chaos in a fourth-order nonlinear parabolic equation S Albeverio, IL Nizhnik Physics Letters A 288 (5-6), 299-304 , 2001 2001.0 Citations: 28
Periodic solutions of second-order differential equations with discontinuous nonlinearity I Nyzhnyk, Krasneeva Journal of Mathematical Sciences 191 (3) , 2013 2013.0 Citations: 14
Differential equations with bistable nonlinearity AM Samoilenko, IL Nizhnik Ukrainian Mathematical Journal 67 (4), 584-624 , 2015 2015.0 Citations: 12
Stable stationary solutions for a reaction–diffusion equation with a multi-stable nonlinearity I Nizhnik Physics Letters A 357 (4-5), 319-322 , 2006 2006.0 Citations: 9
Generalized billiards inside an infinite strip with periodic laws of reflection along the strip's boundaries SA Albeverio, GA Gal'perin, IL Nizhnik, LP Nizhnik Regular and Chaotic Dynamics 10 (3), 285-306 , 2005 2005.0 Citations: 4
Spectral Analysis of Self-Adjoint Momentum Operators with Nonlocal Potentials K Dębowska, I Nizhnik Ukrainian Mathematical Journal, 1-33 , 2025 2025.0 Citations: 1
Spectral analysis of self-adjoint first order differential operators with nonlocal potentials K Dębowska, IL Nizhnik arXiv preprint arXiv:2312.15775 , 2023 2023.0 Citations: 1
Deterministic Diffusion L Nizhnik, I Nizhnik arXiv preprint arXiv:1501.00674 , 2015 2015.0 Citations: 1
Kinklike solutions of fourth-order differential equations with a cubic bistable nonlinearity AM Samoilenko, IL Nizhnik Differential Equations 50 (2), 203-212 , 2014 2014.0 Citations: 1
Stable difference scheme for a nonlinear Klein-Gordon equation IL Nizhnik Ukrainian Mathematical Journal 49 (6), 960-962 , 1997 1997.0 Citations: 1
Deterministic diffusion in dynamical systems with a tiled phase space I Nizhnik arXiv preprint arXiv:2501.00182 , 2024 2024.0