Remarks on Barbashin–Ezeilo Problem on third-order nonlinear differential equations

In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the nth-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution x2(t) is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised.

On the existence of periodic solutions for certain differential equations

In this paper, Lyapunov functions are generated to determine the regions of asymptotic stability of power systems under transient disturbances. With suitable assumptions, the swing equations of the synchronous machines connected to a power system are second- or third-order nonlinear autonomous differential equations. The Lyapunov functions V, employed to determine the domains of asymptotic stability of these nonlinear differential equations, are simple quadratic forms, whose coefficients are chosen so that the Routh-Hurwitz criteria are satisfied for the corresponding linear differential equations. For a synchronous machine swinging against an infinite bus, three typical Lyapunov functions are generated, taking transient saliency and positive-sequence damping into account. The application of Aizerman's method to the same problem leads to a fourth Lyapunov function. The domains of stability given by these Lyapunov functions are compared with the actual stability region obtained by numerical integration. Consideration of one time constant of the prime-mover governor leads to a third-order differential equation. A Lyapunov function is generated, and the stability surface obtained is verified by computing the swing curves numerically for various initial conditions. Finally, it is explained how some of these techniques could be extended for generating Lyapunov functions for multimachine systems.

In this paper, we investigate the asymptotic properties of third-order nonlinear neutral differential equations with mixed nonlinearities using the comparison principle. Our results not only vastly improve upon but also broadly generalize many previously known ones. Examples demonstrating the applicability and efficacy of our results are provided.

Non-linear principal resonance of an orthotropic and magnetoelastic rectangular plate

We perform an all-order resurgence analysis of a quantum field theory renormalon that contributes to an anomalous dimension in six-dimensional scalar ϕ3 theory and is governed by a third-order nonlinear differential equation. We augment the factorially divergent perturbative expansion associated to the renormalon by asymptotic expansions to all instanton orders, in a conjectured and well-tested formula. A distinctive feature of this renormalon singularity is the appearance of logarithmic terms, starting at second-instanton order in the trans-series. To highlight this and to illustrate our methods, we also analyze the trans-series for a closely related second-order nonlinear differential equation that exhibits a similarly resonant structure but lacks logarithmic contributions.

Non-linear oscillations of a third-order differential equation

For a system of differential equations with a cylindrical phase space, we obtain conditions for the existence of several limit cycles of the second kind. These results are applied to phase synchronization systems.

A third-order nonlinear differential equation with two sets of boundary conditions is considered. These boundary value problems arise as boundary layer problems from a model of large scale ocean circulation. Using geometrical techniques from qualitative differential equations, such as Wazewski’s theorem, invariant manifolds, and Lyapunov functions, the existence of solutions for each boundary value problem is given in a uniform way for all positive values of a parameter of the differential equation.

We investigate the Lagrange formulation for the one-dimensional Saint Venant–Exner system. The system describes shallow-water equations with a bed evolution, for which the bedload sediment flux depends on the velocity, Qt,x=Agum,m≥1. In terms of the Lagrange variables, the nonlinear hyperbolic system is reduced to one master third-order nonlinear partial differential equation. We employ Lie’s theory and find the Lie symmetry algebra of this equation. It was found that for an arbitrary parameter m, the master equation possesses four Lie symmetries. However, for m=3, there exists an additional symmetry vector. We calculate a one-dimensional optimal system for the Lie algebra of the equation. We apply the latter for the derivation of invariant functions. The invariants are used to reduce the number of the independent variables and write the master equation into an ordinary differential equation. The latter provides similarity solutions. Finally, we show that the traveling-wave reductions lead to nonlinear maximally symmetric equations which can be linearized. The analytic solution in this case is expressed in closed-form algebraic form.

A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

The magnetic levitation system's control problem is of considerable scientific interest because the system is open-loop unstable and highly non-linear, and the system's parameters are uncertain. In this paper, the system is formulated by a third-order non-linear differential equation. Based on this third-order non-linear system, the system is first linearized by applying the feedback linearization method. Then, the controller is designed based on a global sliding mode technique, which is able to stabilize a linearized system. Experimental results show that, compared with the classic proportional—integral—derivative (PID) controller, the proposed controller provides excellent transient response performance and the system is robust against the non-linear parameter perturbation. Further experiments show that the sampling period is an important parameter to the system robustness. The reason is explained theoretically, and the analysis shows that in the practical global sliding mode control (GSMC) system the smaller the sampling period is, the more robust the controller will be.

This paper considers the control problem of a popular magnetic levitation system, which is open-loop unstable and strongly nonlinear associated with the electromechanical dynamics. The system dynamics is governed by a third-order nonlinear differential equation. The overall controller is designed through a backstepping manner by combining both the robust control and disturbance observer techniques. With the help of nonlinear damping terms, the input-to-state stability (ISS) property of the overall nonlinear control system is proved. Rigorous analysis of the ISS property is given, and experimental results are included to show the excellent position tracking performance of the designed control system.

This paper considers the control problem of a popular magnetic levitation system, which is open-loop unstable and strongly nonlinear associated with the electromechanical dynamics. The system dynamics is governed by a third-order nonlinear differential equation. The overall controller is designed through a backstepping manner by combining both the robust control and disturbance observer techniques. With the help of nonlinear damping terms, the input-to-state stability property of the overall nonlinear control system is proved, which leads to a major academic contribution of construction of a theoretically guaranteed robust nonlinear controller with disturbance observer for the first time in the literature. Experimental results are included to show the excellent position tracking performance of the designed control system.

The circadian rhythm functions as a master clock that regulates many physiological processes in humans including sleep, metabolism, hormone secretion, and neurobehavioral processes. Disruption of the circadian rhythm is known to have negative impacts on health. Light is the strongest circadian stimulus that can be used to regulate the circadian phase. In this paper, we consider the mathematical problem of time-optimal circadian (re)entrainment, i.e., computing the lighting schedule to drive a misaligned circadian phase to a reference circadian phase as quickly as possible. We represent the dynamics of the circadian rhythm using the Jewett-Forger-Kronauer (JFK) model, which is a third-order nonlinear differential equation. The time-optimal circadian entrainment problem has been previously solved in settings that involve either a reduced-order JFK model or open-loop optimal solutions. In this paper, we present (1) a general solution for the time-optimal control problem of fastest entrainment that can be applied to the full order JFK model (2) an evaluation of the impacts of model reduction on the solutions of the time-optimal control problem, and (3) optimal feedback control laws for fastest entrainment for the full order Kronauer model and evaluate their robustness under some modeling errors.