This reproducible steady state may be reached by natural evolution of the system, by artifice, or by a combination of these two.
Examples in everyday life include convection, turbulent flow, cyclones, hurricanes and living organisms.
Less common examples include lasers, Bénard cells, droplet cluster, and the Belousov–Zhabotinsky reaction.
[3] The Hopf decomposition states that dynamical systems can be decomposed into a conservative and a dissipative part; more precisely, it states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and an invariant dissipative set.
Russian-Belgian physical chemist Ilya Prigogine, who coined the term dissipative structure, received the Nobel Prize in Chemistry in 1977 for his pioneering work on these structures, which have dynamical regimes that can be regarded as thermodynamic steady states, and sometimes at least can be described by suitable extremal principles in non-equilibrium thermodynamics.
Two celebrated results from linear thermodynamics are the Onsager reciprocal relations and the principle of minimum entropy production.
Close to equilibrium, one can show the existence of a Lyapunov function which ensures that the entropy tends to a stable maximum.
In chemical systems, this occurs with the presence of autocatalytic reactions, such as in the example of the Brusselator.
Mathematically, this corresponds to a Hopf bifurcation where increasing one of the parameters beyond a certain value leads to limit cycle behavior.
Systems with such dynamic states of matter that arise as the result of irreversible processes are dissipative structures.
Recent research has seen reconsideration of Prigogine's ideas of dissipative structures in relation to biological systems.
A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function
Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems.
In the case of linear invariant systems[clarification needed], this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems[clarification needed].
The framework of dissipative structures as a mechanism to understand the behavior of systems in constant interexchange of energy has been successfully applied on different science fields and applications, as in optics,[12][13] population dynamics and growth[14][15][16] and chemomechanical structures.