Nonlinear suppression using time-delayed controller to excited Van der Pol–Duffing oscillator: analytical solution techniques

Numerous physical phenomena are modeled by nonlinear systems of ordinary or partial differential equations. To examine these systems, it is significant to find out explanations describing the travelling wave phenomenon. On the other hand, the solution of the oscillator equations has received a lot of attention since they are essential in practical mathematics, physics, and engineering demonstrations. The VdPD was reflected as one of the most significant phenomena. The VdPD was used to explain physical, engineering, and even biological complications. This kind of oscillator is a generality of the classic van der Pol oscillator (VdP). Generally, the numerical solution approximation was more complicated than the analytical solution approach to certain problems. The methods of variational iteration [1], HPM [2], Hamiltonian [3], Lindstedt–Poincaré [4], variational [5], parametric expansion [6], max–min methodology [7], iterative harmonic balance [8], and the differential transforms [9] are some of the powerful approaches in analyzing nonlinear oscillator problems that have been performed in the previous works. Using the Melnikov method, Awrejcewicz [10] studied horseshoe chaos in VdPD driven by different periodic forces. Motsa and Sibanda [11] analyzed the linearized method to treat the classical VdPD. By reorganizing the principal equation as a nonlinear eigenvalue problem, they attained correct standards for amplitude and frequency. The chaotic dynamics of the VdP were presented via Adelakun et al. [12] using electronic modeling and hardware employment. Their findings were verified by comparing conformity with the consequences of simulation tools. Khan et al. [13] provided a new approximate approach to solving VdPD problem. Only two homotopy series components, the Laplace transforms and the Padé approximates, are used in the proposed system. The vibrational motion of many dynamical models with different degrees of freedom is examined [14‐19]. The controlling systems are analyzed by Lagrange’s formula to generate the fundamental equations of motion and are solved using the multiple scale approach. The solvability criteria are investigated by eliminating the secular terms, and the various resonance situations are examined. The modulation equations are examined numerically to analyze the frequency response curves and determine the stability and instability areas.

Delay differential equations, whose current state oscillation is reliant on preceding states, can be used to model the diversity of the technological and medical sciences. The Hopf bifurcation [20] is studied using the center of manifold theory [21], which is a computationally efficient methodology in the qualitative treatment of delay differential equation bifurcations. Time delay is a vital topic in dynamic vibration controller performances, where the presence of time delay in the controller round might be the foremost purpose of the organization’s disappointment via destabilizing the control loop. In current history, differential equations, incorporating delayed bearing, have gained considerable attention. These equations have proven to be successful in the modeling of a wide range of applications in technology and engineering [22, 23]. Saeed et al. [24] addressed six different time-delayed controllers and their benefits as well as disadvantages in controlling a parameterized stimulated system of nonlinear fluctuations. Studies that are both quantitative and computational, however, demonstrated that the cubic-acceleration controller is the best choice. Saeed et al. [25] developed a time-delayed position-velocity regulator to overpower a nonlinear system vibration. The influence of specific organizer parameters on the system vibration characteristics was examined. Furthermore, numerical approximations of the analytical consequences were achieved. For the first time, Saeed et al. [26] investigated the use of a nonlinear fundamental resonant oscillator to reduce the major parameterized disturbance. The controlling loop time delay was included in the system under examination. Additionally, the loop delay collection mechanism that either enhances the control representation or destabilizes the network movement has been thoroughly explained.

As known, nonlinear equations are required in the majority of real-life and technical applications. These equations are either functional, differential, integral, or integro-differential. The exact solutions to these equations are extremely hard to find. As a consequence, the branching of numerical solutions in several ways becomes critical. Because analytic analyses are more useful in many situations than numerical ones, perturbation techniques have evolved in various forms, ranging from the traditional strained parameter to the modification methods such as those based on several time scales. The fundamental principle of perturbation techniques is to turn a collection of first-order equations from nonlinear to linear equations. The Poincaré–Lindstedt technique was used by He et al. [27] to attain an approximate bounded solution for the hybrid Rayleigh VdPD. The estimated solution and the fourth-order Runge–Kutta approach were found to be similar. In fact, all perturbation methods necessitate the presence of a small parameter in the equation being studied. Subsequently, the problem becomes relatively constrained in the absence of such a parameter. He [28] went on to use a new perturbation method, independent of such a small parameter. According to this strategy, a modest artificial embedding parameter can be placed by \(\delta\) where \(\delta \in [0,1]\). In case \(\delta = 0\), the differential equation of zero order must have an exact solution. In order to attain a valid restricted approximate solution to the parameterized Duffing equation (DE), Moatimid [29] employed an expanded frequency parameter and a combination of HPM and Laplace transform. Ghaleb et al. [30] used a similar approach to get a circumscribed estimated explanation of the cubic-quintic VdPD. Recent works related to the existing manuscript were found through Refs. [31‐33]. In the case of the autonomous system, they also achieved a linearized stability profile at the equilibrium points. The achieved outcomes are considered to be novel and original, in which the used strategy is applied to a particular dynamical system.

In accordance with the significance of the aforementioned aspects, the analysis of the VdPD has potential applications in engineering, physics, communication theory, and biology. It has been discussed in numerous papers on a variety of topics. Therefore, this paper analyzed the excited VdPD with position and velocity delays as follows:where all parameters included in Eq. (1) are listed at the beginning of the paper. It must be noted that the time delay parameter belongs to the interval \([ - \tau ,\,0)\).

The numerical solution (NS) of Eq. (1) can be gained utilizing the Runge–Kutta technique of fourth order (RK4) in the presence of the following data:

This solution has been drawn in parts of Fig. 1 according to the various values of the natural frequency \(\omega\) and the delayed parameter \(\tau\). A closer look at the waves included in portions (a) and (b) of this diagram shows that these curves are starting at the highest larger amplitude of y-axis when \(t = 0\), and then a decrease of this amplitude is observed as time goes on. Consequently, the performance of these waves has stable manner since they have a steady behavior at the end of the considered time interval. Moreover, parts (c) and (d) are plotted to describe the relationship between the NS and its first derivative for the same considered values of \(\omega\) and \(\tau\). The sketched curves represent the phase plane diagrams of the obtained results, and they have the form of oriented spiral curves inward toward a single point, which confirms the stability of the numerical results.

In Eq. (1), the term \(\mu (1 - y^{2} )\dot{y}\) is termed as damping (dissipation) term. In the absence of the last term, Eq. (1) is turned to the DE which has much significance in the background of many nonlinear systems. To crystallize this work, the remainder of the article is structured as follows: Sect. 2 is depicted to introduce the analytic solution, based on the HPM of Eq. (1). Additionally, its subsection is produced to present a modification of HPM to attain a more accurate solution. The previous solutions are numerically confirmed. The obtained solutions are plotted to reveal the influence of the delayed parameter and the natural frequency of motion. Moreover, the phase plane diagrams describing the stability are presented and discussed. Section 3 and its subsections investigate two resonance cases, in addition to the examination of the super- and sub-harmonic resonance cases. Section 4 presents the results of this work.

The controlling equation of motion (1) is a nonlinear equation with a periodic coefficient. In reality, it has no specific solution. As a result, it will be analyzed by the traditional HPM. He [28] successfully utilized his perturbation approach to reach a periodic solution for numerous oscillators. However, his method fails for the damping oscillator problems. In analyzing Eq. (1), it is appropriate to present the subsequent original equations:

The procedure of the HPM is mainly based on the following homotopy equation:

Along with this approach, the dependent variable may be required in the traditional procedure by way of:

To yield the solution, one inserts the expansion (4) into the homotopy Eq. (3). It follows that the analytical exact explanation of the zero-order equation is specified by

Accordingly, one finds

The first-order problem of the Homotopy Eq. (3) might be written by way of:

With the aid of Eqs. (5)–(8) into Eq. (8), one gets

The uniform valid phrase is usually obtained by removing the secular terms. The coefficients of the functions \(\cos \,\omega t\) and \(\sin \,\omega t\) should be ignored at all stages for this goal.

At this stage, the following initial conditions are considered:

The bounded solution at the principal step is specified by

As a result, what follows is the constrained suitable estimation of the fundamental equation provided in Eq. (1) as:

The time history of the previous solution (14) and the corresponding phase plane, at two different values of the natural frequency \(\omega\), are drawn in Fig. 2a, b, respectively. These parts are graphed according to the previous data when \(A = 1\).

An examination of Fig. 2a indicates that the behavior of the obtained solution is periodic and the number of waves increases with the decrease of \(\omega\) values. Therefore, the wavelength of these waves decreases, while the amplitudes remain stationary. The phase plane diagrams at the considered values of \(\omega\) are represented by the included closed curves of Fig. 2b. These curves indicate that the given solution in Eq. (14) has a stable behavior and that it is free of chaos.

The comparison between the obtained approximate solution (AS) in Eq. (14) and the numerical solution (NS) of the fundamental Eq. (1) is graphed when \(\omega = 3.252\) in portions a–c of Fig. 3 at \(\tau ( = 0.7,\,\,0.4,\,\,{\text{and}}\,\,0.1)\), correspondingly. It is evident that there is no deviation in the AS through the change of the standard values of the decelerating parameter \(\tau\), while the variation of the NS with this parameter is evident through the curves drawn in the parts of this figure. The deviation between the two solutions is very large. The reason is due to the fact that Eq. (1) depends on \(\tau\), while the AS is independent of \(\tau\) as seen in Eq. (14).

In the reality, excitation frequency occurs in groups. A resonance happens when the frequencies of external loads precisely or substantially match the frequency response of a system, and the oscillations that result have just a large amplitude. In potential implementations, it has both positive and negative influences. Resonating, a technology used to inhibit the oscillation of a framework during excitations, is one of the beneficial implications of resonance. The excitations cause destruction of bridges and airplanes owing to the effects of resonance.

The objective of several investigators in nonlinear differential equations is to arrive at theoretical and numerical solutions. Analyzing an approximate solution could be accomplished in a number of approaches in reality. The goal of this work is to look at a time-delayed control system for suppressing nonlinear VdPD. In a one-degree-of-freedom organization, the governed equation of motion can be reduced to an ordinary nonlinear differential equation. Despite the fact that the elimination of the secular terms produces a fine solution, unfortunately, the HPM results in a uniform approximate solution, which does not match the numerical solution provided by RK4 approach. As a result, the damping behavior of the needed solution is treated by a modified HPM. To validate the latter solution, numerical verification is used to back up the new analytic approximate solution. The comparison of the several solutions reveals a high level of regularity, confirming the technique's extreme accuracy. In addition, the mathematical analysis and solution for resonance cases are derived. These analyses include both the super-harmonic resonance situation (\(\Omega \cong \omega /3\)) and the sub-harmonic resonance circumstance (\(\Omega \cong 3\omega\)). The graphical representations of the analytic approximate solution and the different modulation equations are presented to illustrate the impact of various restrictions on the inspected motion. It is noted that the motion is stable and unrestricted of chaos. Since the employed method is applied to the investigated dynamical system, the results are considered to be new and original.