The occurrence of complex dynamic behavior (i.e., bifurcation processes, strange and chaotic attractors) in autonomous space-invariant cellular neural networks (CNNs) is investigated. Firstly, some sufficient conditions for the instability of CNNs are provided; then some classes of unstable template are identified. Finally, it is shown that unstable CNNs often exhibit complex dynamics and for a case study the most significant bifurcation processes are described. It is worth noting that most CNN implementations exploit space-invariant templates and so far no example of complex dynamics has been shown in autonomous space-invariant CNNs.