On local attractivity of a coupled non-linear differential equation

Question

Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(t) &= w + a_{21}\sin(x_1(t)-x_2(t)) + a_{23}\sin(x_3(t)-x_2(t))\\ \dot{x}_3(t) &= w + a_{31}\sin(x_1(t)-x_3(t)) + a_{32}\sin(x_2(t)-x_3(t)), \end{align} $t\ge 0$, $x_i(0)\in\mathbb{R}$, $i=1,2,3$, where $v$, $w\in\mathbb{R}$, $a_{ij}\in\mathbb{R}$, and $a_{12},a_{13},a_{21},a_{31}\ne 0$.

I'm studying the properties of this dynamical system. So far, I've managed to show that if $a_{21}=a_{31}$ then the system has an invariant trajectory given by $\bar{x}(t)=(x_1(t),x_2(t),x_2(t))$, $t\ge 0$ (that is, $x_2(t)=x_3(t)$, $t\ge 0$).

Open problem: Is the invariant trajectory $\bar{x}(t)$ locally attractive?

Numerical simulations seem to suggest that the answer is in the affirmative. However the proof of this conjecture does not seem trivial to me. This could be due to the fact that I'm rather new on this kind of (local) stability problems. So I would be enormously grateful in hearing any comment/criticism/suggestion from you. Also, pointers to the literature are very welcome.

Answer 1

Since the RHS is $2 \pi$-periodic in all variables, one can consider it on the three-dimensional torus $(\mathbb{R}/2 \pi \mathbb{Z})^3$.

Assume $a_{21} = a_{31}$. Then the two-dimensional torus $$ T := \{\, (x_1, x_2, x_2): x_1, x_2 \in \mathbb{R}/2 \pi \mathbb{Z} \,\} $$ is an invariant submanifold. To investigate its stability, I propose to use the function $u := x_2 - x_3$.

By subtracting the third equation from the second and performing some transformations we obtain (if my computations are O.K.) $$ \tag{1} u'(t) = - 2 a_{21} \cos(x_1(t) - \tfrac{1}{2}(x_2(t) + x_3(t))) \sin(\tfrac{1}{2}u(t)) - (a_{23} + a_{32}) \sin(u(t)). $$ The zero stationary solution of (1) corresponds to $T$. Formally linearizing (1) along the zero solution we obtain $$ \tag{2} y'(t) = - a_{21} \cos(x_1(t) - \tfrac{1}{2}(x_2(t) + x_3(t))) y(t) - (a_{23} + a_{32}) y(t). $$ In the general (multidimensional) case a sufficient condition for the global asymptotic stability of the zero solution of (1) is the existence of $c \ge 1$ and $\lambda > 0$ such that $$ \tag{3} \lVert \Phi(t, s) \rVert \le c \exp(- \lambda (t -s)), \quad s \le t, $$ where $\Phi(t,s)$ denotes the solution operator (transition or Cauchy matrix). (Caveat: in the multidimensional case, for $\xi'(t) = A(t) \xi(t)$, the property that the real parts of the eigenvalues of $A(t)$ are, for all $t \in \mathbb{R}$, less than some negative number is not a sufficient condition for (3).) However, in the one-dimensional case we can make use of the comparison property of solutions to an ODE: $-1 \le \cos(x_1(t) - \tfrac{1}{2}(x_2(t) + x_3(t))) \le 1$, so we can compare equation (2) with the autonomous equation $$ z'(t) = (\lvert a_{21} \rvert - (a_{23} + a_{32})) z'(t). $$ From the properties of the cosine function it follows that all the estimates appearing above are uniform both in $t$ and w.r.t. any solution $(x_1(\cdot), x_2(\cdot), x_3(\cdot))$.

So, if $$ \lvert a_{21} \rvert < a_{23} + a_{32} $$ then the invariant torus $T$ is uniformly asymptotically (exponentially) stable.