Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions $X = (x_1, x_2, ..., x_N)$, we have
$X'' = A*X$
How can this system become chaotic by introducing an extra term? For instance, would it become chaotic by adding a higher-order term like $f_{(x_1)}$ to the equation of the first oscillator, such as $x_1^2$ or $x_1^4$? Or, do I need to add nonlinear coupling?
linearly coupled anharmonic oscillators exhibit chaotic dynamics, a simple example studied by Steep, Louw, and Villet is
$$\ddot{x}_1=-A x_1-ax_1^3-cx_2$$ $$\ddot{x}_2=-Ax_2-ax_2^3-cx_1$$
You can get chaos with a cubic term for $x_1$. In applied dyanmical systems, there has been considerable interest in last decade to study systems of the following form: $\ddot{x_1}=-ax_1^3+\epsilon(x_1-x_2)$
$\ddot{x_2}=bx_2+\epsilon(x_2-x_1)$, where usually $\epsilon<<1$, and rest of coefficients are $O(1)$.
Physically, this corresponds to weakly & linearly coupled oscillators , and presence of a strong nonlinearity (a PURELY nonlinear spring attached to first). The solutions of this class of equations have been seen to be chaotic for a subset of parameters.
See for example: Energy pumping in nonlinear mechanical oscillators:Part 1 by Gendelman et. al. Journal of Applied Mechanics.
