Duffing and Van der Pol Oscillators are typical examples of nonlinear dynamic systems and thus we will use them as a reference to test the implemented controllers. Both oscillators are good examples of periodically forced oscillators with non-linear elasticity. A Duffing oscillator can be represented by the mathematical model shown in Equation (1); on the other hand a Van der Pol Oscillator mathematical model is shown in Equation (2).
A forced or driven oscillator means that a driving function of is added to Equations (1) and (2). The updated mathematical model will become as shown in Equations (3) and (4) representing the Duffing and the Van der Pol Oscillators respectively.
In our application, we apply an actuator for the oscillators and this yields the updated mathematical model that will be used in our calculations. Equation (5) represents a driven Duffing oscillator with an actuator. Equation (7) on the other hand, represents a driven Van der Pol Oscillator again with an actuator. Mathematical models of the forced Duffing and Van der Pol systems are shown respectively.
From Equation (5) the state space was deduced to be as shown below:
From Equation (7) the state space was deduced to be as shown below:
where, , and, are all given constants  . is the given actuator. Equation (9) represents the mathematical model of the actuator.
In a previous work  , we presented a technique to control a chaotic Duffing Oscillators using recursive back stepping Control  . In this paper, we will continue on the work that was presented earlier by showing a solution for the Van der Pol oscillator as well. The work presented in this paper will also deal with the mathematical model when an actuator is added as shown in Equations (6) and (8). The next section will list and describe some of the related works that have been done in this field. The following section will describe our proposed method with some of the generated results. The paper is then ended with a conclusion and a future work.
2. Related Work
There are numerous applications for Duffing and Van der Pol Oscillators. For example, the Duffing oscillator has been used to do chirp signal detection  . It is also widely used in the signal communication domain such as in the secure communication field as shown in  and weak signal detection as shown in  and  . It has also found its ways to marine applications such as ship propeller blade number recognition as shown by Wang et al. in  . Van der Pol oscillator has had less application examples than the Duffing oscillator nevertheless it still found its way into numerous domains such as in the medical field where was used in the modelling of the cardiac pulse as shown in  . Another example is the use of the Van der Pol oscillator in the modeling of the dust density wave fields as shown in  .
There has been some work done on the control of a Duffing Oscillator. Kuo et al.  used a fuzzy sliding controller. The control rules were based on the Lyapunov stability theorem, simulation results proved that it can successfully control the system even with the presence of chaos. Alexander Jimenez-Triana et al.  also presented chaos control for a Duffing system but this time using impulsive parametric perturbations. The presented approach has been established based on Melnikov’s method  and the authors confirmed the success of the proposed method using numerical simulations.
The authors in  investigated the dynamic characteristics of the Van der Pol system with added delay. The authors found that Hopf bifurcation occurs from trivial equilibrium when the delay passes through critical values. The authors then found the critical values and their relationship with the system parameters. The authors proved their results using numerical results. Yang  et al. studied chaos control in a Van der Pol system with nonlinear force and two forcing excitations. The authors proved their results using numeral simulation. The authors concluded that chaotic motions are controllable by adjusting the phase difference and the amplitude of the second excitation force. Lastly, Van der Pol system control was also done using bifurcation such as the work that was done by M. Xiao et al.  .
3. Proposed Method and Chaos Analysis
3.1. Duffing Analysis
Equation (5) represents the forced Duffing oscillator alongside an actuator. Let the error be defined as the. This means that.
Let be defined as shown in Equation (10) thus can be defined as shown in Equation (11) and finally and are defined as shown in Equations (12) and (13) respectively. and are constants representing tunable parameters.
Substituting and in Equation (13) will yield
This means that
Rearranging Equation (5) will yield the following:
, substituting the variables and rearranging will yield to Equation (14) as shown below.
Now if we suppose that and we follow the same procedure, will be generated as shown in Equation (15).
The system performance was captured at the values below as shown in Table 1.
The following figures show the results of the presented control system. Figure 1 shows a Duffing Actuator Position on Velocity with a stable trajectory for Duffing equation with sinusoidal drive using the Grapher application. On the other hand, Figure 2 shows the Duffing Actuator Position on Velocity Unstable Chaotic trajectory for Duffing equation with sinusoidal drive also using the Grapher application. In both cases, it is assumed that is and is. Figure 3 shows a Duffing Stable solution at 25 in poles with comparison to a reference sinusoidal drive. Figure 4 on the other hand shows a Duffing uncontrollable system when compared to a reference sinusoidal drive. Looking at the position trajectory in Figure 5, it can be noticed that it is uncontrollable position with a sinusoidal drive using the Grapher application, it is also assumed here that that is and is. Finally Figure 6, shows an unstable velocity in comparison to the reference sinusoidal drive.
Table 1. System Performance captured at these specific values.
Figure 1. Duffing Actuator Position on Velocity Stable.
Figure 2. Duffing Actuator Position on Velocity Unstable.
Figure 3. Duffing Stable at 25.
Figure 4. Duffing uncontrollable.
3.2. Van der Pol Analysis
The same principle and logic is applied again to the Van der Pol analysis. Equation (16) and Equation (17) represent the solution when is assumed to be
Figure 5. Duffing uncontrollable position.
Figure 6. Duffing Unstable Velocity.
Table 2 shows the system best performance.
Using the same flow for showing the results for the Van der Pol system. Figures 7-12 shows the same series of plots as the Duffing system. Figure 7 shows the Van der Pol Actuator Position vs. Velocity with a stable trajectory for a Van der Pol equation with sinusoidal drive. On the other hand, Figure 8 shows Van der Pol Actuator Position vs. Velocity Unstable Chaotic trajectory for a Van der Pol equation with sinusoidal drive. Again, in both cases it is assumed that is and is. Figure 9 shows a Van der Pol Stable solution at 25 in poles with comparison to a reference sinusoidal drive. Figure 10 on the other hand, shows a Van der Pol uncontrollable system when compared to a reference sinusoidal drive at pole 7.5. Looking at the position trajectory in Figure 11, it can be noticed that it is uncontrollable position with a sinusoidal drive, it is also assumed here that that is and is. Finally Figure 12, shows an unstable velocity in comparison to the reference sinusoidal drive.
Table 2. System Performance captured at these specific values.
Figure 7. Van der Pol Actuator Position on Velocity Stable.
Figure 8. Van der Pol Actuator Position on Velocity Unstable.
Figure 9. Van der Pol Stable at 25.
Figure 10. Van der Pol Uncontrollable.
Figure 11. Van der Pol Uncontrollable Position.
Figure 12. Van der Pol Unstable Velocity.
3.3. Energy Calculations
The potential and kinetic energies are studied in this section. The same analytical process is used as sections “a” and “b” earlier. The kinetic and the potential energy equations are shown in Equation (18) and (19) prospectively.
Using the same analysis as before the kinetic energy solution for both Duffing and Van der Pol oscillator, respectively, can be expressed as shown in Equations (20) and (21).
Using similar analogy, the potential energy for a Duffing and Van der Pol oscillator, respectively, can be expressed as shown in Equations (22) and (23).
The dissipation energy for both the Duffing and the Van der Pol oscillators can be expressed as shown in Equation (24).
This will yield dissipation energy solution as shown in Equations (25) and (26) representing the Van der Pol and Duffing respectively.
The system best performance was then recorded at the values below:
Table 3 shows the Duffing forced;
Table 4 shows the Van der Pol forced;
Table 5 shows the Duffing unforced values;
Table 6 shows the Van der Pol unforced values.
Using these solutions the results are shown in Figures 13-15. Figure 13 shows
Table 3. System Performance captured at these specific values.
Table 4. System Performance captured at these specific values.
Table 5. System Performance captured at these specific values.
Table 6. System Performance captured at these specific values.
Figure 13. Duffing Energy change rate.
Figure 14. Forced Van der Pol.
the energy exchange rate in the unforced Duffing oscillator. This figure shows the kinetic, mechanical, and the potential energies as well as the dissipated power. Figure 14 on the other hand, shows the energy exchange in a forced Van der Pol oscillator. Again this figure shows the mechanical, input energy, and the dissipated energy. Finally, Figure 15 shows the energy exchange in an unforced Van der Pol oscillator. Figure 15 shows the kinetic, mechanical, and the potential energies as well as the dissipated power.
Figure 15. Unforced Van der Pol Energy.
In this paper, we presented a novel nonlinear control method that was applied to forced Duffing and Van der Pol oscillators that were experiencing chaotic behavior to a prescribed performance. The oscillators had an actuator applied to them. We also presented the energy exchange in forced Duffing and Van der Pol oscillators. The paper illustrated the usefulness of the presented method in the unstable areas. The presented controllers achieved two objectives: we first stabilized both the Duffing oscillator and the Van der Pol oscillators. Secondly, we presented the transient performance of the system. Robustness can be added to the system as a future work. This can be achieved by incorporating states estimator, or parameters estimator or even both. These added estimators can be integrated into the design by introducing more virtual control constraints and changing the corresponding Lyapunov function. As an additional future work, we would like to incorporate the effort of this work into another work that we did earlier and more specifically to the photovoltaic system control with the presence of an electric vehicle and a home load as we showed in   and  .