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The harmonic balance method is broadly employed for analyzing and predicting the periodic steady-state solution. Most of the traditional methods in the literature do not guarantee global optimality. Due to its nonconvexity, it is a complex task to find a global solution to the harmonic balance problem in the frequency domain. A convex relaxation in the form of semidefinite programming has attracted attention since its introduction because it yields a global solution in most cases. This paper introduces a novel optimization-based approach to predict periodic solution by determining the Fourier series coefficients with high accuracy. Unlike the other commonly used methods, the proposed approach is completely independent of initial conditions. In our proposed method, the nonlinear constraints composed of time-dependent trigonometric functions are converted into nonlinear algebraic polynomial equations. Then, nonlinear unknowns are convexified through moment-sum of squares approach. However, computing a global solution costs a higher runtime. Our approach is validated through small examples which contain only polynomial nonlinearities. In all cases, the Mosek solver shows a better performance in comparison with SDPT3 and SeDuMi solvers. The proposed method shows high computational cost as a result of an increase in the positive semidefinite matrix size, which can depend on the number of harmonics and the degree of nonlinearity.

We expose the chaotic attractors of time-reversed nonlinear system, further implement its behavior on electronic circuit, and apply the pragmatical asymptotically stability theory to strictly prove that the adaptive synchronization of given master and slave systems with uncertain parameters can be achieved. In this paper, the variety chaotic motions of time-reversed Lorentz system are investigated through Lyapunov exponents, phase portraits, and bifurcation diagrams. For further applying the complex signal in secure communication and file encryption, we construct the circuit to show the similar chaotic signal of time-reversed Lorentz system. In addition, pragmatical asymptotically stability theorem and an assumption of equal probability for ergodic initial conditions (Ge et al., 1999, Ge and Yu, 2000, and Matsushima, 1972) are proposed to strictly prove that adaptive control can be accomplished successfully. The current scheme of adaptive control—by traditional Lyapunov stability theorem and Barbalat lemma, which are used to prove the error vector—approaches zero, as time approaches infinity. However, the core question—why the estimated or given parameters also approach to the uncertain parameters—remains without answer. By the new stability theory, those estimated parameters can be proved approaching the uncertain values strictly, and the simulation results are shown in this paper.