In this paper the first Lyapunov coefficient of the emerging periodic solution in the Hopf bifurcation is established. A change of coordinates is introduced to eliminate some quadratic and cubic terms in the dynamic equations and the center manifold theory is used to reduce the dynamics to ...
Moreover, the stationary covariance matrix Σ for the state can be computed by solving the following Lyapunov equation $$A{{\Sigma }}+{{\Sigma }}{A}^{T}+D{D}^{T}=0,$$ where D is the positive semidefinite matrix satisfying \(D{D}^{T}=S{{{\rm{diag}}}({{\Lambda }}\bar{...
The Lyapunov function is designed to prove that the proposed controller can reach a convergence state within finite time. Finally, the proposed control method is compared with PID control, fuzzy PID control, linear sliding mode control, and super spiral control combined with a disturbance observer ...
Therefore, Λ(0) is the maximum Lyapunov exponent of the isolated system which is equal to 0 for a periodic dynamics, leading to the vanishing of the critical value ν* as well as of the threshold for complete synchronization. In turn, this would imply that intermediate clusters could not ...
Can be easily proved by considering the Lyapunov function $$\begin{aligned} G= & {} \frac{1}{2}(E-E^*)^2+\frac{m_1}{2}(I_s-I_s^*)^2+\frac{m_2}{2}(I_a-I_a^*)^2+\frac{m_3}{2}(Q-Q^*)^2\nonumber \\&+\frac{m_4}{2}(A-A^*)^2+\frac{m_5}{2}(N-...
The boundedness of the solution of model (3-1) and the Lyapunov–LaSalle invariance principle of the functional differential equation show that, if R0≤1, then E0 is globally asymptoticallyrobust (stable). Theorem 2 For any time delay factor τ≥0, when R0>1, the risk equilibrium point E...
We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous differential system of a special form as t ↑ ω, ω≤ + ∞. For this system, the coefficient matrix of a differential system of the first approximation has almost Jordan form with triangular ...
first() - last() - d() calculating a metric related to the "disorder" or "inefficiency" of the data. It takes two arguments, src (the input data series) and itv (the length of the calculation). The function sorts the input data in descending order and then iterates over the sorted...
Aisagaliev, S.A., Analiz i sintez avtonomnykh nelineinykh sistem avtomaticheskogo upravleniya na osnove vtorogo methoda Lyapunova (Analysis and Design of the Automatic Control Systems by the Second Method of Lyapunov), Alma-Ata: Nauka, 1980. Google Scholar Gelig, A.Kh., Dinamika im...
Using the Lyapunov–Schmidt reduction it was shown in58 that the bifurcations of a single NLS are pitchfork. For the case of BEC the coefficient of the cubic nonlinear term is positive. Thus the pitchfork bifurcations are supercritical where the solution curves turn to right. It is straightforwar...