Phase Self-Stabilization

Abstract

If x = x₀(t) is the solution of the system of equations EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadI % hacaGGVaGaamizaiaadshacqGH9aqpcaWGybGaaiikaiaadIhacaGG % OaGaamiDaiaacMcacaGGPaGaaiilaiaadIhacqGHHjIUcaGGOaGaam % iEamaaCaaaleqabaGaaiikaiaaigdacaGGPaaaaOGaaiilaiaac6ca % caGGUaGaaiOlaiaacYcacaWG4bWaaWbaaSqabeaacaGGOaGaamOBai % aacMcaaaGccaGGPaGaaiilaiaadIfacqGHHjIUcaGGOaGaamiwamaa % CaaaleqabaGaaiikaiaaigdacaGGPaaaaOGaaiilaiaac6cacaGGUa % GaaiOlaiaacYcacaWGybWaaWbaaSqabeaacaGGOaGaamOBaiaacMca % aaGccaGGPaaaaa!5EB5!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$dx/dt = X(x(t)),x \equiv ({x^{(1)}},...,{x^{(n)}}),X \equiv ({X^{(1)}},...,{X^{(n)}})$$ with the initial condition x₀ (t₀) = a, then the condition x₀ (t + γ) = a evidently corresponds to the solution x = x₀(t + γ). The set of motions of an autonomous dynamic system permits arbitrary shifts in time. Solutions of the form x₀ (t + γ) for arbitrary fixed values of γ are combined by the curve [x₀(θ)] in {x} space by saying that the motions with a given trajectory [x₀(θ)] differ only by a constant phase difference.† This concept is generalized to motion along different (nearby) trajectories. Denoting the image point of the motion x(t) on the curve [x₀(θ)] by x₀(t + γ(t)), γ(t) is called the phase difference between the motions x(t) and x₀(t) [1, 2]. Phase considerations make it possible to explain properties of the motion associated with shifts in time, thereby easing the analysis of complicated systems.