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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.

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    $\begingroup$ Perhaps you know this already, but dynamical systems similar to yours are sometimes known as "Kuramoto oscillators" en.wikipedia.org/wiki/Kuramoto_model and there is a rather large literature on them. I am not an expert but at first glance this paper ieeexplore.ieee.org/document/7525284 seems discuss your particular system where all of the $a_{ij}$ are equal. $\endgroup$ – j.c. Feb 28 '18 at 19:11
  • $\begingroup$ @j.c.: Yes, I knew this. Thanks for the reference: it could be interesting! $\endgroup$ – Ludwig Feb 28 '18 at 19:21
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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.

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  • $\begingroup$ Thanks for your answer! Your argument is ingenious and looks technically sound to me (I need to carefully check the computations though)! I've also the impression that it could be "generalized" to provide conditions for stability in more complex cases (i.e. for systems with dimension larger than 3) $\endgroup$ – Ludwig Mar 26 '18 at 13:17
  • $\begingroup$ Could you please elaborate a little more about the comparison property to solutions to ODE that you mentioned? Do you have a reference for this? Thanks! $\endgroup$ – Ludwig Mar 27 '18 at 14:32
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    $\begingroup$ I mean a theorem stating that if $\varphi$ [resp. $\psi$] is a solution of $x'= f(t, x)$ [resp. $x' = g(t,x)$] with $x(t_0) = x_0$ and $f(t,x) < g(t,x)$ for all $t$ then $\varphi(t) < \psi(t)$ for $t > t_0$ as long as both solutions are defined. Its proof is straightforward, but when we relax $<$ to $\le$ the situation changes: either one has to assume something additional (e.g. Lipschitz) or there are counterexamples (see math.stackexchange.com/questions/912468/… and math.stackexchange.com/questions/158332/…). $\endgroup$ – user539887 Mar 28 '18 at 9:16
  • $\begingroup$ Many thanks for the clarification! A last curiosity: In all my simulations I've noticed that (local) stability is also promoted by choosing $v$ and $w$ very different (i.e. choosing $|v-w|$ very large); hence I was wondering whether some sufficient conditions for local stability that involve the terms $v$ and $w$ could be derived. By chance, do you have any idea? $\endgroup$ – Ludwig Apr 3 '18 at 5:13
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    $\begingroup$ No, I don't have any idea. Incidentally, see my answer to your other question Behavior of a non-linear differential equation. $\endgroup$ – user539887 Apr 8 '18 at 10:45

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