Chaos can be observed in simple nonlinear Hamiltonian systems. This is a dynamical system governed by Hamilton's equations where the energy is conserved, such as a physical pendulum or double pendulum. The phase space portrait of the trajectories of this kind of system can, even with few degrees of freedom, be very complicated. The phase space flow fulfils Liouville's theorem, which states that phase space volume is conserved. Another type of chaotic dynamics can arise in non-autonomous systems, like the Duffing oscillator, where a simple nonlinear system is influenced by an external periodic force. A third kind of chaotic system is nonlinear dissipative systems, such as the Lorenz (1963) model, which has only three degrees of freedom. The Lorenz model was derived from the set of ordinary differential equations describing development of wave amplitudes in the spectral representation of Rayleigh–Bernard convective flow. The Lorenz model is equivalent to a spectral truncation where only the first three wave numbers are represented. The phase space portrait of dissipative systems is different from that of Hamiltonian systems because the energy dissipation implies a shrinking of phase space volume. The dynamics of such a system is described in phase space by strange attractors. Strange attractors are sets in phase space of states un which are invariant with respect to the dynamical equation. This means that an initial state un(0) belonging to the attractor will develop along a trajectory which will stay within the attractor.