Stability in the sense of Lyapunov is associated with the equilibrium point. Moreover, an oscillator cannot oscillate forever, unless the energy lost while oscillating or in the resistances  is supplied back which is also not a part of such a notion or stability in the Lyapunov sense, 

Perhaps you would like to have a rotating frame of reference in which case it would be interesting but am not sure how such ideas are handled.

To me Lyapunov's ideas talk mechanics.

For my opinion, the "stability of an oscillator" refers to the stability of the oscillating FREQUENCY. There is no oscillator which provides a signal with a frequency which is really constant. Each oscillator output provides a signal having a certain phase noise (short-term stabiliy) as well as a certain degree for a long-term stabiliy.

The oscillation in an oscillator is a limit cycle, that is , aa asymptotic periodic solution which does not depend on initial conditions ( amplitude and frequency do not depend on initail conditions). This allows to fix it with the same parameters each time the system is operated  ( oscillators in TV, radio etc,)  It is a generated in the presence of some nonlinearity ( typically , a saturation) acting on a linear network.

Stability here typically refers to the stability of the limit cycle ( the oscillation) in the sense that any perturbation of it is asymptotically removed. The response of Sudha is right,  in my opinion, Lyapunov´s method,  through Lyap. functions,  is related to stability of equilibrium points , not to that  limit cycles.

If the nonlinearity is separable from the linear one ( for instance a saturation of an amplifier in tamdem, but separated in the lopo,  with a linear third order system), the describing function method can be useful. Take the decribing function - minus inverse of the characteristic locus of the saturation- and take the hodograph of the linear part frequency response together in the Nyquist plane, Then , check if the locuses intersect ( the intersection on the linear frequency response and on describing function give apprimated values of  the" ocillation amplitude" and of the   " first harmonic  oscillation frequency", respectively. There is a graphic test of stability , instability as well to be performed in this description which appears in books of nonlinear systems.

In the case that the oscillation is observed in an oscillatior device, for instance,the the oscillation is clearly stable since it does not vanish.

Hello, Trusna! According to automatic control theory, after stability of an oscillator we can mean both stability of its frequency and stability of its oscillates (in terms of equilibrium point).

What exactly we mean depends on field of your interests.

If we talk about electronic oscillator to be applied as a clock oscillator or a model (template) of ideal frequency we have to be concerned about stability of frequency. Indeed, here we can separate long-term instabilities and short-term one. Long term instabilities are slow changes in oscillator frequency over time (e.g., minutes, hours, or days), generally due to temperature changes and/or oscillator aging. For good oscillators, this instability is measured in parts per million (ppm). Parts per million is a similar to describing the instability in terms of percentage change in oscillator frequency. A more direct way of expressing “parts per million” is “Hz per MHz”—in other words the amount of frequency change in Hz, divided by the oscillator frequency expressed in MHz. 

The short-term instabilities of oscillators are commonly referred to as phase noise—a result of having imperfect resonators (elemets of circuit). With phase noise, the random process  has very small magnitude, but changes very rapidly (e.g., milliseconds or microseconds). This is equivalent to narrow-band frequency modulation, and the result is a spreading of the oscillator signal spectrum. 

But, for example, if we talk about mechanical oscillates of some part of load of electric drive (cargo handed to bogie starts to oscillate due to moving of bogie) then we interested in the second definition of oscillator stability. In this case we can describe oscillation by means of equations (that is nonlinear, unfortunately) and can apply any feasible criteria of stability. For example, it could be criterias of Lyapunov, Nyquist, et al.

I hope I get the core of your question clear. Best wishes! 

Thanks all for your valuable answers. So stability of an oscillaor basically means the stability of it's oscillations or the limit cycle or attractor. As in system theory, the limit cycle is ragarded as marginal stability, so can we say that an oscillator works at marginally stable conditions?? Can we show this by taking a +ve semi-definite Lyapunov function for a particular system?? What are the available methods to detect stable limit cycle(or attractors) for a system, specifically for Affine systems?? 

Trusna,

A limit cycle CANNOT be regarded as a marginally stable oscillation. In linear systems you will have such a situation if you get two poles on the imaginary axis (and all the rest in the lwft half plane). Notice that this is STRUCTURALLY UNSTABLE because any small perturbation of the parameters, and your undamped oascillation is gone. On the other hand, a limit cycle is STRUCTURALLY STABLE, because even in the presence of parameter perturbations the limit cycle will exist. 

Dear Trusna,

there are two aspects in your question. Firstly, the term "stability of an oscillator" is vague and inaccurate. Better we can say of "stability of an oscillation". From the mathematical perspective, oscillation may be considered as a periodic solution or as a closed trajectory (a limit cycle in the state space). As it was noticed previously, a nonlinear system can demonstrate a multistable behavior, i.e., it may produce several different stable oscillations, each having its own basin of attraction. For analyzing limit cycles an auxilairy discrete time map (a Poincare map) is oftenly constructed in the space of trajectories. The initial data for a limit cycle is

obtained as a fixed point of this map or of its iterations. Thus, we can associate with an oscillator an original continuous time equation and an auxiliary Poincare discrete time equation (or discrete time equations in a multistability case).

Secondly, about the stability.

I am strongly against the opinion that Lyapunov stability refers to equilibria only. (The fundamental work of Lyapunov written in 1895 is accessible in English and you can check it if you wish.) Lyapunov stability and asymptotical Lyapunov stability refer to any chosen solution of a differential or discrete time equation, and to a periodic solution in particular. If you consider an oscillation as periodic solution, you can investigate its Lyapunov stability using standard stability technique.

However, in most cases they consider not a Lyapunov, but an orbital stability, i.e. stability in the state of trajectories. (Orbital stability does not necessary imply Lyapunov stability of a corresponding periodic solution.) To prove orbital stability of an initial (differential) system they oftenly consider an equilibrium of a Poincare (discrete time) map an prove its asymptotical Lyapunov stability (in the sense of discrete time equations). That is how it usually works, not so simple I guess.

Hi everybody. In Khalil's Nonlinear Systems 3rd edition section 8.4, this matter is somehow discussed. By the end of the section there's the definition of a "orbitally stable" periodic solution, the same that Dr. Alexander Churilov has refered in the second part of his answer.

Dear Trusha Meher,

You got some excellent answers.

I would like to separate Lyapunov approach to stability analysis, which everybody uses, from Lyapunov’s original Theorem of Stability. The approach requires to fit to the system and appropriate (energy-like?) Lyapunov function V(x) and use its derivative “along the trajectories” Vdot(x) in order to see how the Lyapunov function and so, the trajectories, develop. Lyapunov’s theorem requires the derivative to be negative definite. In this case, the derivative must ultimately be zero and, for a negative definite function, Vdot(x)=0 is equivalent with x=0  or with asymptotic stability.

However, in most cases other than class-room examples, the derivative is at most negative semidefinite and I must warn you that here the analysis and the conclusions are more complex. In this case, the system still ends at Vdot=0, yet now, one must see the meaning of that. If the system happens to be globally asymptotically stable (with one equilibrium point), one may still reach this conclusion.

However, V (x) could also be a function of, say, 20 variables, x1 to x20, while the derivative could be  Vdot(x)=-x1^2. The conclusion still is that the system ends at x1=0, yet does not say anything about all the other 19 variables.

On the other hand, if your system is function of x and y and the derivative comes out to be something like Vdot(x) =-(x^2+y^2-1), then Vdot is positive for x^2+y^21. This leads to the conclusion that trajectories that start within x^2+y^21 come towards the origin. This must lead to the conclusion that all trajectories must end at within x^2+y^2=1, which then is a stable limit cycle.

Much work with semidefinite Lyapunov derivative has been done by LaSalle in his 1976-1980 works, even though they have remained pretty much unknown, even in the textbooks that were mentioned here. In order to give it some more popularity, along with some new results, I wrote the paper

I. Barkana: "Defending the Beauty of the Invariance Principle," International Journal of Control, Vol. 87, No. 1, pp. 186-206, 2014 (Published On-Line 06 Sep 2013), DOI:10.1080/00207179.2013.826385.

This is not necessarily about limit cycles in particular, yet the stability analysis approach fits your question pretty well.

Best regards,

Itzhak

Thanks a lot every one... I really got some excellent answers... :) :) 

it seems that you look for stability of limit cycle behavior. you can look for stability of limit cycle, it's somehow complicate but it can be found from different approach. An interesting one is Floquet theory.