Can a linear system be chaotic? A chaotic system is a system in which infinitesimal perturbations of a parameter can result in large changes in the behavior of the system.  I thought it is not possible for a linear system to exhibit chaotic behavior, but some recent papers (e.g., this paper) indicate that infinite-dimensional linear systems can be chaotic.  It seems that any system that includes unbound particles would be infinite dimensional, in that wavefunctions of those particles would belong to an infinite-dimensional Hilbert space.  So is it actually possible for a linear QM system to be chaotic?
 A: tl;dr–  Yes, any function, including chaotic functions, can be decomposed into an infinite linear system, therefore infinite linear systems can be chaotic.

A trivial example would be
$$
f\left(x\right)
~=~
\sum_{\forall i}{
a_i \, \delta\left(x-p_i\right)
}
\,,$$
where $\delta\left(x\right)$ is the Dirac delta function and there's one amplitude/position tuple, $\left<a_i,\,p_i\right> ,$ for each discriminable $x ,$ $x_i .$  In other words, just think of an infinite-resolution piece-wise function.
Then $f\left(x\right)$ is non-chaotic in the special case that $\require{cancel} \left| a_i - a_{i-1} \right| ~\cancel{\!\! \gg \!\!}~ \left| p_i - p_{i-1} \right| ~~ \forall i \,,$ though $f\left(x\right)$ is almost certainly chaotic.  In other words, an infinite-resolution piecewise function is non-chaotic if we select the values for each point to differ by not much more than the distance between the points, but since this is an extremely contrived special case, almost all infinite-resolution piece-wise functions are chaotic.
Since this is a linear combination of trivial atoms (I selected the Dirac delta since it's pretty easy to reproduce in most numeric systems, to help keep this explanation more general), yup, linear systems of infinitely many components can be chaotic.
Notes:


*

*Any function $f\left(x\right)$ can be decomposed into such an infinite-resolution piece-wise function.

*Chaotic behavior is observer-subjective.  This appears in the above where we define chaotic-quality as being a function of $\require{cancel} \cancel{\!\! \gg \!\!} ,$ which is a fuzzy qualifier.
A: There is certainly a definition issue if one accepts that “linear” equations describe systems that do not interact and “nonlinear” equations describe ones that do.  Under this definition “chaos” can only arise in interacting systems described by “nonlinear” equations.  
However, “linear” equations can be solved by matrix methods, and assuming the problem can be represented with a square matrix, the system will have a characteristic polynomial associated with the matrix.  Finding the roots of the polynomial is how one would solve the system of equations.  Characteristic polynomials can be difficult to solve for roots especially in high degree systems but can be solved using iterative methods such as Newtonian methods.  It is in solving for the roots of a polynomial that fractal/chaotic behavior can be observed, and quite beautifully at that (see link below).  
So chaos can exist in linear systems just not necessarily in a direct way.
https://pi.math.cornell.edu/~hubbard/NewtonInventiones.pdf
Update:  There are some good introductory papers on infinite dimension linear systems and chaos. One which deals with this topic for biological models is available here http://www.scielo.org.za/pdf/sajs/v104n5-6/a0610406.pdf
