By Conti M., Gazzola F.

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**Extra resources for Existence of ground states and free-boundary problems for the prescribed mean-curvature equation**

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250000000. Hence the ﬁeld lines of the originar system are linear unstable, and the extended system is Jacobi unstable. The Jacobi stability depends on the variation of the parameters. We shall discuss further the impact of the change of the parameters γ and θ on the structural behavior of the extended SODE (3)–(4). Considering the parameter θ variable within the interval (0,1) with γ ﬁxed, we get the following results regarding the linear and the Jacobi stability. 2. Let γ = 4 (the limit cycle value).

It was proved that for γ = 4, one obtains a limit cycle and that the SODE exhibits a Hopf bifurcation; it provides a model with negative feedback (inhibition) as shown in the following diagram excitation excitation inhibition x1 −−−−−−−→ x2 −−−−−−−→ x3 −−−−−−−→ x1 . 0. In practice are used iterated pulses with initial value z0 and zn = δ(e−nτ /tc − 1) z0 e−nτ /tc + . Then the asymptotic value (for n → ∞) is 1 − eτ /tc z∞ = δ/(eτ /tc − 1). The response γ or γ0 and the time constant tc are refered as network parameters — which are inherent properties of the network, and δ and τ are referred as stimulation parameters — which are the stimulation pulse amplitude and stimulation period, respectively.

Chern, Sur la geometrie d’un syst`eme d’equations diﬀerentialles du second ordre, Bull. Sci. Mat. 63 (1939), 206–212. 9. D. Kosambi, Parallelism and path space, Math. Zeitschrift 37 (1933), 608– 618. 10. R. Nicola and V. Balan, Jacobi stability for dynamical systems with applications to biology, Proc. of the 3-rd International Colloquium “Mathematics in Engineering and Numerical Physics” (MENP-3), 7–9 October 2004, Bucharest, Romania, BSG Proceedings 12, Geometry Balkan Press, Bucharest, 195–201.

### Existence of ground states and free-boundary problems for the prescribed mean-curvature equation by Conti M., Gazzola F.

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