ABSTRACT
Here, we consider another ODE model (132) for N = 1, and, without loss of generality, we mainly restrict to m = 2. We show that (132) provides us with similar countable families of various patterns. Moreover, we claim that the solution set of (132) is equivalent to that obtained earlier for non-Lipschitz nonlinearities. Now, solutions F (y) of (132) are not compactly supported and exhibit an oscillatory exponential decay by the linearized operator:
F (4) = −F + ... =⇒ F (y) = c1e−
(√ 3 2 y + c2
) , y → +∞, (179)
where c1,2 ∈ IR are arbitrary constants. As above, it comprises a 2D bundle.
Figures 1.42 and 1.43 show a few typical patterns, which we are already familiar with. It is important to notice that, in Figure 1.42, by watching the behavior of small negative solutions close to the origin y = 0, there are two different patterns that can be classified as F+2,2,+2, and the second one is denoted by F ∗+2,2,+2. This shows again that the number of intersections with equilibria ±1 and 0 are not enough for a complete pattern description (in fact, this emphasizes that a homotopy approach using the hodograph plane is not applicable to equations such as (132)). It is seen there that F ∗+2,2,+2(y) exhibits a more non-monotone structure for y ∈ (−4, 4) than F+2,2,+2(y), so that the derivative F ∗
