Weakly coupled oscillators & 3-regular graph

After visiting my Ph.D adviser Dr Ermentrout at Pitt on Aug, 19,   I have been trapped into the following problems.

Problem 1:  To find a non-trivial solution for weakly coupled oscillators on a 3-regular graph for the coupling function H(θ) = sin(θ).

To solve this problem, one may start with a simpler problem.

Problem 2:  To find a non-trivial stable solution for weakly coupled oscillators on a 3-regular graph with a Hamiltonian cycle via the same H.

It is actually  a ring of oscillators  with multiple-node coupling (each is  coupled with 2-nearest neighbors  + 1 additional nodes).  Using homotopy method, we may  be able to arrive at a solution that is homotopy to the  nontrivial one on the ring: (0, 2π/N, …, 2(N-1) π/N ):

(a)  The nodes can be denoted as  1,2,…, N as they present  in the Hamiltonian cycle;

(b)  θ’ =  ω + H(θ_k-1 - θ_k) + H(θ_k+1 - θ_k) + λ H(θ_p(k) – θ_k)  where p(k) is the 3^rd coupling node  of k in the cycle;

(c) Let λ change from 0  to 1 to find out out whether there is any solution at λ=1 which is homotopy to  (0, 2π/N, …, 2(N-1) π/N ). 

A solution should exist when p(k) satisfies certain conditions:
(a) p(k) = (k+h) mod N;
(b) N=2m > 10;
(c) 1 < h < N/4.