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For mechanical and structural systems, one can often use the energy as the Lyapunov function. In electrical power systems, some methods have been used to determine Lyapunov functions and one can define feedback loops based on this. For Hamiltonian systems, the Hamiltonian can be taken as the Lyapunov function.

However, how does one define the physical meaning of a 'Lyapunov function' in dissipative systems? Say for example, an ecosystem forest growth over centuries with rain and fire as the other variables?

Sunny Marella
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2 Answers2

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The intuition behind the Lyapunov function in dissipative systems has been extensively discussed by Ilya Prigogine in his theroy of irreversible (non-equilibrium) thermodynamics, dissipative structures and complex systems as well as by Hermann Haken in his theory of synergetics.

The mathematical intuition or physical meaning of the Lyapunov function is indeed identical with the Hamiltonian, and sometimes used by some authors as Hamiltonian. But particularly in the theory of dissipative systems where systems described by order parameters and the eigenvalues of the linear part (control parameters), the Lyapunov function gains an important meaning as a potential function of the so called master modes (potential relates terminologically to potential energy in classical thermodynamics) that in a very comprehensive way lets us understand phase transition phenomena, such as critical fluctuations, bifurcations and symmetry breaks near equlibrium while the system non-linear and dissipative.

The potential function includes also some significant characteristics of the attractors of the system, such as oscillatory and stability etc - this is also because they involve the linear eigenvalue properties as well as the non-linear properties of the system.

Dependent on the system (and field of application chem, bio...) the Lyapunov function can help to understand also oscillatory phenomena and pattern fomration (spacial and temporal).

In pattern recognition the situation reverses, there a Lyapunov function is defined which has the attractors as the paterns against which we recognize.

Also in Chaos theory the Lyapunov function gains special meaning, where can be shown that certain properties of the exponent are prerequisite to deterministic Chaos. Beyond the deterministic theory the Lyapunov function can be also found with similar interpretations (but distinguished) in the stochastic theory of dissipative systems when for instance applying Fokker-Planck equations or Langevin equations.

In fractal theory we can trace relations between the Lyapunov function and the fractal dimmension - but this statement I raise very carefully.

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What I can suggest you is to read through the interdisciplinary work of both Prigogine and Haken.

With regard to your example, the one ecosystem can be regarded for instance as a stable near equlibrium but nonlinear dissipative system where the potential function after a phase transtion from a so called "V" phenomenlogically seen turns to a "W" which is again stable. The other system may have control parameters changed in a way that the "V" turns to a saddle and instable.

Hope this is good answer!

A posterior

yes correct your comment, that was writing fast. It is the Lyapunov function I am talking about. However, beware there is a deep relation between the Lyapunov function and the Lyapunov exponent. This is why I wrote "but this statement I raise very carefully". When a system in deterministic chaos, the tarjectories behaviour can be traced by the iterative (often fractal process) taking the Lyapunov exponent. One can show, at least phenomenologically, that the Lyapunov exponent encapsulates (entropic) information from the modal trajectories (i.e. in eigenspace) that is identical with information that in a possibly existing corresponding deterministic model is supported by the Lyapunov function. The trivial case for intance to exercise would be to identify the Lyapunov function (Hamiltonian, potential) of a population growth model (that can under certain control parameters like 3 instable modes turn chaotic) and then the Lyapunov exponent (trivially related to its eigenvalues).

al-Hwarizmi
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  • I think you meant 'Lyapunov function' and not 'Lyapunov exponent'-which is a indicator of chaos and separation of trajectories. – Sunny Marella Jun 19 '13 at 05:41
  • yaes see extension. – al-Hwarizmi Jun 19 '13 at 12:47
  • In the same light of defining terms based on physical systems, how would one classify 'hyperbolic' and 'nonhyperbolic' fixed points? By saying that a Hopf point if a nonhyperbolic fixed point, what would this mean in a 'REAL' physical system? NOTE: Acc. to my book, Hopf bifurcation was observed in a model of the centrifugal governor(the mechanical system used to control the flow of fuel depending on the speed rpm of the engine). The book doesn't give the physical meaning of the STRUCTURAL CHANGE as a result of the Hopf bifurcation unfortunately. – Sunny Marella Jun 27 '13 at 10:18
  • @Sunny Marella it wont be possible to answer this question in a comprehensive way here. This will require the extensive discussion of the different cases. One SOS imagination could be to think about LF as a surface: LF V(x,y) for instance while x and y span the phase plane. You can cut parallel V axis (orthogonal xy-plane) and discuss the different cases. But a well discussion you will also find in Hermann Haken "Synergetics - An Introduction" Chapter 5.2 discussing the critical points and trajectories in the phase plane. I hope this gives good answer to all your questions. – al-Hwarizmi Jun 27 '13 at 12:33
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For systems like damped pendula, the apparent Hamiltonian - kinetic plus potential but excluding energy lost as e.g. heat - remains a Lyapunov function, with $\dot H<0$ because apparently energy is lost, being converted into unmonitored heat energy via friction.

In general, there is no one physical interpretation of all Lyapunov functions for all systems; there may be an interpretation for any one given system.

not all wrong
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  • firmly disagree, there is indeed a universal interpretation of the Lyapunov function possible that is through reinterpretation of entropy to negentropy and Shanon information. Practically this approach can reinterprete the meaning of the Lyapunov function as a potential (e.g. a surface) where it descibes the system behaviour versus information gain. Since the concept of inforamtion can be applied to any such deterministic system, there can be given a universal interpretation of the Lyapunov function; see Jaynes principle: https://en.wikipedia.org/wiki/Principle_of_maximum_entropy – al-Hwarizmi Jun 27 '13 at 10:59
  • I don't consider that a physical interpretation for several reasons, but I suspect the debate would be unproductive so will leave it at that. – not all wrong Jun 27 '13 at 21:23