   One might think that SD, Chaos\(_{te}\) or Chaos\(_{\lambda}\) could be sufficient for defining chaos, but these characterizations run into problems from simple counterexamples. For instance, consider a discrete dynamical system with \(S = [0, \infty)\), the absolute value as a metric (i.e., as the function that defines the distance between two points) on \(\mathbf{R}\), and a mapping \(f: [(0, \infty) \rightarrow [0,\infty)\), \(f(x) = cx\), where \(c \gt 1\). In this dynamical system, all neighboring trajectories diverge exponentially fast, but all accelerate off to infinity. However, chaotic dynamics is usually characterized as being confined to some attractor—a strange attractor (see sec. 5.1 below) in the case of dissipative systems, the energy surface in the case of Hamiltonian systems. This confinement need not be due to physical walls of some container. If, in the case of Hamiltonian chaos, the dynamics is confined to an energy surface (by the action of a force like gravity), this surface could be spatially unbounded. So at the very least some additional conditions are needed (e.g., that guarantee trajectories in state space are dense). 