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?



Making a system of second-order ODEs chaotic
linearly coupled anharmonic oscillators exhibit chaotic dynamics, a simple example studied by [Steep, Louw, and Villet](http://www.publish.csiro.au/?act=view_file&file_id=PH870587.pdf) is


$$\ddot{x}\_1=-A x\_1-ax\_1^3-cx\_2$$
$$\ddot{x}\_2=-Ax\_2-ax\_2^3-cx\_1$$


