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Some analytical criteria for local activity of three-port CNN with four state variables: analysis and applications. (English) Zbl 1057.37076

Summary: This paper presents some analytical criteria for local activity principle in reaction-diffusion Cellular Nonlinear Network (CNN) cells [L. O. Chua, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, No. 10, 2219–2425 (1997; Zbl 0901.68138), IEEE Trans. Circuits Syst., I, Fundam. Theory Appl. 46, No. 1, 71–82 (1999; Zbl 0948.92002)] that have four local state variables with three ports. As a first application, a cellular nonlinear network model of tumor growth and immune surveillance against cancer (GISAC) is discussed, which has cells defined by the Lefever-Erneaux equations, representing the densities of alive and dead cancer cells, as well as the number of free and bound cytotoxic cells, per unit volume. Bifurcation diagrams of the GISAC CNN provide possible explanations for the mechanism of cancer diffusion, control, and elimination. Numerical simulations show that oscillatory patterns and convergent patterns (representing cancer diffusion and elimination, respectively) may emerge if selected cell parameters are located nearby or on the edge of the chaos domain. As a second application, a smoothed Chua’s oscillator circuit (SCC) CNN with three ports is studied, for which the original prototype was introduced by Chua as a dual-layer two-dimensional reaction-diffusion CNN with three state variables and two ports. Bifurcation diagrams of the SCC CNN are computed, which only demonstrate active unstable domains and edges of chaos. Numerical simulations show that evolution of patterns of the state variables of the SCC CNN can exhibit divergence, periodicity, and chaos; and the second and the fourth state variables of the SCC CNNs may exhibit generalized synchronization. These results demonstrate once again Chua’s assertion that a wide spectrum of complex behaviors may exist if the corresponding CNN cell parameters are chosen in or nearby the edge of chaos.

MSC:

37N25 Dynamical systems in biology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G99 Local and nonlocal bifurcation theory for dynamical systems
92C15 Developmental biology, pattern formation
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