User talk:Artinanita

Tavakolpour-Lyapunov Theorem
Instability of an equilibrium state in a dynamical system with periodic response (e.g. oscillators) can be affirmed if a PD function V(x) is found such that

V(x) = 0           If         x = 0						       				        (1)

V(x) > 0           If         x ≠ 0										        (2)

〈 dV(x)/dt 〉_2π > 0       If     x  is considered as a fictitious periodic function (i.e. x = εcos(t))			(3)

where ε is the amplitude of oscillation at the vicinity of the equilibrium point and 〈.〉 is averaging operator which can be defined for periodic functions i.e. g as

〈g〉= 1/2π ∮_2π g(s)ds. Besides, the proposed theorem can also be applied to the stability problems of periodic systems as well if just the third criterion is modified as

〈 dV(x)/dt 〉_2π < 0     If     x    is considered as a fictitious periodic function (i.e. x = εcos(t))			     (4)

Eq. (4) implies that if the average value of V ̇(x) is negative the system under consideration will be dissipative in general and thus, the equilibrium state at the origin is stable.

The proof of this theorem is given in [1] in which it is shown that the real parts of eigenvalues of a linear system (poles of a dynamical system) directly appear in 〈 dV(x)/dt )〉_2π meaning that the study of 〈 dV(x)/dt 〉_2π provides important information regarding instability as well as the stability of a dynamical system around the origin.

Remark: The use of the average power instead of the instantaneous power is also a conventional technique in different fields of engineering such as acoustics, which can justify the third criterion of the theorem as well. Indeed, the Tavakolpour-Lyapunov theorem is more flexible than the original Lyapunov theory. In other words, every Lyapunov function can satisfy this theorem; however, based on the presented theorem other positive definite functions can also be found to prove instability of a fixed point in the phase plane, which do not satisfy the conventional Lyapunov theory. Indeed, finding such PD functions about an equilibrium state can prove the instability (or stability) of the fixed point based on this theorem as well. More evidence to support the mentioned claim can be found in [1].

Reference:

[1] A.R. Tavakolpour-Saleh, Sh. Zare, "An averaging-based Lyapunov technique to design thermal oscillators: A case study on free piston Stirling engines", Energy 189 (2019) 116127.