We remark here that, for the Jordan closed curve, we have the following Jordan theorem [3]: An arbitrary Jordan closed curve must divide a plane into two regions, and one of the regions is bounded and the another is unbounded.
If (a<0), then an attracting invariant closed curve bifurcates from (E_{2}) for (h^>0).
Here we denote by (mathcal{D}(Gamma)) the open bounded region bounded by the Jordan closed curve Γ.
The bounded region is called the interior and the another is called the outside of the Jordan closed curve.
Moreover, (k < 0), and thus an attracting invariant closed curve bifurcates from the fixed point for (d>bar{d}^).
In this paper, we assume that Γ is a smooth and convex Jordan closed curve in (mathbb{R}^{2}) [1 3].
From (15) in Section 3 we know that the centered surround system (S^{(2)} {P,varGamma }) exists for any smooth and convex Jordan closed curve Γ. Theorem 1 implies the following interesting corollary.
In the convex geometry, a well-known isoperimetric inequality can be expressed as follows: If Γ is a smooth Jordan closed curve, then we have biglvert {D varGamma )}bigrvert leqslantfrac{|varGamma |^{2}}{4pi}.
Surface features are classified into three groups: open curves, closed curves and singularity features.
Manual correction by a biologist verified that DCL cells were not in contact to each other and EVL segmented membranes were closed curves.
Algaba, A., Merino, M., Fernandez-Sanchez, F., Rodriguez-Luis, A.J. "Closed curves of global bifurcations in Chua's equation: a mechanism for their formation," International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, vol.13, no.3, pp.609-616.
