Stationary distribution and extinction of a stochastic HIV/AIDS model with nonlinear incidence rate

1.   Introduction

In recent years, applications of controllers using the saturation phenomenon developed to end system's vibrations. Many researchers studied this phenomenon in different dynamical systems. Oueini et al. ^[1] recommended an active control for a cantilever beam by adding a vibration damper. The flexible structure with the controller has been presented analytically using multiple scale method at two-to-one autoparametric resonances. Pai et al. ^[2] improved the performance of NSC and the linear position feedback algorithm analytically, numerically and experimentally. From this study they showed that NSC is efficient to decrease the steady-state vibrations. Oueini and Nayfeh ^[3] a control strategy for the excited beam by cubic velocity feedback under a principal parametric resonance is suggested. The analysis detected that this controller reduced the amplitude of the response. Pai and Schulz ^[4] introduced a controller connected to a system with quadratic terms depended on saturation occurrence. They controlled the oscillation of the system via PZT patches as actuators and sensor. Pai et al. ^[5] considered some absorbers depended on saturation phenomena designed in support of declining the vibrations of a plate model. Ashour and Nayfeh ^[6] used an absorber based on saturation phenomena for lessening vibrations of the model. In favor of raising the action of the saturation control they determine the frequency of excitation together with the system. Jun et al. ^[7] illustrated the saturation controller to the structure by processing typical PZT patches. Depended on analytical solutions these authors showed that the nonlinear absorber (NSC) was globally stable, as distinct from a linear model where growing feedback gain value might instead go in front to instability. However, for a nonlinear absorber, the power need will be greater than that for the linear system. Jun et al. ^[8] presented a nonlinear saturation controller for decreasing vibrations of a self-excited plant. The control mechanism performed via combination the absorber among the plant applying quadratic nonlinearity. Xu et al. ^[9] proposed the influence of a time-delay in a structure depending of a linear beam with NSC absorber. The analytical results based on the multiple scales method revealed much more complex dynamics for the system. Presence of the time-delay widened or decreased the frequency bandwidth of effective vibration suppuration.

EL-Sayed ^[10] presented the purpose delay positive position feedback control (DPPF) in support of reduction vibrations in the Van der Pol oscillators system with external forces. The effectiveness of vibrations suppuration by DPPF is tested for selected parameters to obtain the stability condition of the model. Warminski et al. ^[11] concentrated on appliance of special controllers for a flexible beam with MFC actuators. Mathematical solutions for the beam with NSC obtained using perturbation technique. The best controllers for reduction the vibration for the system are NSC and PPF controllers. Saeed et al. ^[12] illustrated time-delay saturation controller to decrease oscillations of nonlinear beam. Hamed, and Elagan ^[13] considered effeteness of NSC control algorithms for huge oscillation of the beam structure. Hamed and Amer ^[14] used NSC to decrease the oscillation amplitude of a composite beam. Kamel et al. ^[15] proposed a magnetically levitated body with a NSC to decrease the horizontal vibration simulated by a nonlinear differential equation. Omidi and Mahmoodi ^[16] for nonlinear oscillation decreasing, they performed the Positive Position Feedback (PPF), the Integral Resonant Control (IRC) and Nonlinear Integral Positive Position Feedback (NIPPF). They solved analytically the system using multiple scales method. Conclusions show that controller construct has mainly job in suppression performance.

Numerous main papers are studied the vibration absorbers on various nonlinear systems. Linear and nonlinear dynamic vibration absorbers have been employed to suppress the primary resonance vibrations of a weakly nonlinear oscillator systems having cubic nonlinearity subjected to external excitations under primary resonance conditions ^{[17,18,19,20]}. The method of multiple scales is used to obtain the averaged equations that determine the amplitudes and phases of the first-order approximate solutions to the vibrations of the primary nonlinear oscillator and nonlinear absorber. The nonlinear absorber can effectively suppress the amplitude of primary resonance response and eliminate saddle-node bifurcations occurring in the frequency-response curves of the primary nonlinear oscillator. Numerical results are given to show the effectiveness of the nonlinear absorber for suppressing nonlinear vibrations of the primary nonlinear oscillator under primary resonance conditions. Moreover, different control approaches are investigated and confirmed to decrease the unsafe vibrations which are made in different nonlinear systems ^{[21,22,23,24,25]}. Furthermore, Ji et al. ^[26] studied the dynamic behavior of a quadratic nonlinear oscillator involving time delay under two-to-one resonances of two Hopf bifurcations. Also, Ji ^[27] investigated the secondary resonance response of a time-delayed quadratic nonlinear oscillator after the trivial equilibrium of the system loses its stability via two-to-one resonant Hopf bifurcations. Zhang and Li ^[28] investigated the global bifurcations and chaotic dynamics of a nonlinear oscillator with two-degree-of-freedom. The fast and slow modes may exist simultaneously, and the chaotic motions were found by using numerical simulation. Chaos in beams has been widely investigated recently. With the Galerkin method, the method of multiple scales and numerical simulations, Zhang ^[29] investigated the chaotic motion and its control for the nonlinear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. By means of the Galerkin procedure, normal form theory and a global perturbation method. Zhang et al. ^[30] investigated the bifurcations and chaotic dynamics of a simply support symmetric cross-ply composite laminated piezoelectric rectangular plate, which are simultaneously forced by the transverse and in-plane excitations and the excitation loaded by the piezoelectric layers. Hao et al. ^[31] studied the complicated nonlinear dynamics of a functionally graded material cantilever rectangular plate subjected to the transverse excitation in thermal environment. Zhang et al. ^[32] analyzed nonlinear dynamic responses of cantilever rectangular laminates with external excitation. Nonlinear forced vibration characteristics of carbon nanotube reinforced composite plates were studied by Guo and Zhang ^[33]. Lu et al. ^[34] proposed a robust control method for the vibration suppression of composite laminated cantilever rectangular plates subjected to the aerodynamic force in hygrothermal environment. Recently, Siriguleng et al. ^[35] studied the dynamic response of a rotating laminated composite blade in the case of 2:1 internal resonance and the author reported the interesting saturation phenomena in the rotating composite blade.

The key goal of this research is to provide a general guideline for removing high vibrations of the framework by using a suitable control technique. The nonlinear saturation control (NSC) considered for active suppression of high vibration for the nonlinear dynamical system which presented in ^[16]. The analytical result received by using perturbation technique process near simultaneous internal and primary resonance case. The comprehensive mathematical solutions, frequency response equations (FREs), and stability analysis with NSC process are obtained using the perturbation method. Numerical solution and effect of all parameters on the vibrating system and NSC system are plotted and reported. The MATLAB software is used to simulate the impact of various parameters and the controller on the system. The validations of the analysis's time history and frequency response curves (FRCs), as well as the numerical results, were satisfied by comparing them. Before and after providing the controller in the simultaneous internal and primary resonance case, the system is numerically and graphically examined. The simulation outcomes give that the NSC method is found to be the most effective at eliminating high vibrations and making the system more stable. Finally, numerical results are obtained that show an outstanding agreement per the analytical results.

2.   Governing equations and approximate solutions

In this section, the model for the nonlinear dynamics of a cantilever beam is governed by a nonlinear partial differential equation. The primary resonant mode of the cantilever beam is considered not to be involved in an internal resonance with other modes of the system. Single-mode discretization approach results in nonlinear differential equation of ^[16] yields:

To perform the multiple-time-scale procedure, system parameters are scaled, such that:

So, the equation motion of the cantilever beam obtained from ^[16] is modified by using NSC process as follows:

where ${\eta _s}$ and ${\eta _c}$ are linear damping coefficients of the main system and NSC, $f$ and $\Omega$ are the forcing amplitude and the frequency of the system, ${\omega _s}$ and ${\omega _c}$ are natural frequencies of the main system and NSC, $\alpha, \beta$ and $\gamma$ are nonlinear factors, ${\gamma _1}$ is control signal gain, ${\gamma _2}$ is feedback signal gain and $\varepsilon$ is a small perturbation value.

Figure 1 shows connection of NSC system to the main system in a closed figure which describing Eqs (2.2) and (2.3).

2.1.   Perturbation analysis

In this section, the perturbation manner ^[36,37] is useful to seek the approximate solutions of Eqs (2.2) and (2.3) as known:

The time derivatives of the Eqs (2.2) and (2.3) can be written in terms of two time-scales as:

Inserting Eqs (2.4) to (2.6) in Eqs (2.2) and (2.3), and compare coefficients of similar power of $\varepsilon$, we achieve:

The solutions of Eqs (2.7) and (2.8) are:

Where, ${A_1}({T_0}, {T_1}), \, \, {A_2}({T_0}, {T_1})$ are complex functions in ${T_0}$ and ${T_1}$, $cc.$ locate for the complex conjugate of the preceding terms. Substitute Eqs (2.11) and (2.12) interested in Eqs (2.9) and (2.10), we obtain the following solutions:

Wherever the over bar indicates the complex conjugate function. Before we proceed to the next step of the solution, we have to establish the possible resonance cases at Eqs (2.13) and (2.14) which be primary resonance ($\Omega \cong {\omega _s}$), internal resonance (${\omega _s} \cong 2{\omega _c}$), and simultaneous resonance ($\Omega \cong {\omega _s}, \, \, {\omega _s} \cong 2{\omega _c}$) that is the worst one. Therefore, this case is investigated by introducing the two parameters ${\sigma _1}$and ${\sigma _2}$to describe quantitatively the closeness of $\Omega$ and ${\omega _c}$ to ${\omega _s}$as follows:

Inserting Eq (2.15) into the small-divisor and secular terms of Eqs (2.9) and (2.10), we get the following solvability conditions:

For analyzing Eqs (2.16) and (2.17), we rewrite ${A_1}({T_1})$ and ${A_2}({T_1})$ in polar form as follows:

Wherever, ${a_1}$ and ${a_2}$ are the steady-state amplitudes of the system and${\beta _1}$, ${\beta _2}$ are the phases of the two motions. Applying Eq (2.18) into Eqs (2.16) and (2.17) and sorting out the real and imaginary parts, we acquire the following set of first-order differential equations.

Where, ${\theta _1} = {\sigma _1}{T_1} - {\beta _1}$, and ${\theta _2} = {\sigma _2}{T_1} + {\beta _1} - 2{\beta _2}$.

Equations (2.19) to (2.22) form the system amplitude-phase modulating equations.

3.   Stability analysis

The variations of amplitudes and phases of a periodically excited system are zero at steady state. So, the algebraic equations that govern the steady state vibrations of the considered structure can be found from Eqs (2.19) to (2.22) by setting${\dot a_1} = {\dot a_2} = {\dot \theta _1} = {\dot \theta _2} = 0$, yields

From Eqs (3.1) to (3.4) we have the following cases:

(I)${a_1} \ne 0, {a_2} = 0$ (II)${a_2} \ne 0, {a_1} = 0$ and (III)${a_1} \ne 0, {a_2} \ne 0$.

In support of the helpful case (III), Squaring Eqs (3.1) and (3.2), after that count the squared results together, similarly do the same for Eqs (3.3) and (3.4), next we contain the successive frequency response equations:

To perform stability criteria, we assume ${a_{10}}, \, \, {a_{20}}, \, \, {\theta _{10}}$ and ${\theta _{20}}$ are the solutions of Eqs (3.1) to (3.4), and to observe the performance of small perturbations ${a_{11}}, \, \, {a_{21}}, \, \, {\theta _{11}}$ and ${\theta _{21}}$ from the steady state solution ${a_{10}}, \, \, {a_{20}}, \, \, {\theta _{10}}$ and ${\theta _{20}}$, we let

Substituting Eq (3.7) into Eqs (2.19) to (2.22) and expanding for little parameters ${a_{11}}, \, \, {a_{21}}, \, \, {\theta _{11}}$ and ${\theta _{21}}$ with carry on linear terms only. We compact the linear system that is equivalent to Eqs (2.19) to (2.22) at the equilibrium point (${a_{10}}, \, \, {a_{20}}, \, \, {\theta _{10}}, \, {\theta _{20}}$):

Wherever the over matrix is the system Jacobian matrix, and its coefficients are prearranged in the Appendix. One could be able to get the following eigenvalue equations:

By expanding determinant at Eq (3.9), yields

where $\lambda$ is the eigenvalue of the matrix, ${\zeta _1}$, ${\zeta _2}$, ${\zeta _3}$ and ${\zeta _4}$ are the coefficients of Eq (3.10) known in the Appendix. By applying Routh-Hurwitz criterion for stability, we find the necessary and sufficient conditions to be stable solution are:

4.   Numerical simulation results

In this part, the original system Eqs (2.2) and (2.3) are plotted via Runge-Kutta algorithm (by MATLAB 7.14 (R2012a) package) numerically. Figure 2 shows the responses of the model without control process at primary resonance $\Omega \cong {\omega _s}$ (one of worse resonance cases) and zero initial. This shape appears that, amplitude of the system $x$ regarding 829% of the excitation amplitude$\, f$. Figure 3 demonstrate response and phase-plane for the nonlinear dynamical system with NSC at the simultaneous resonance case $\Omega \cong {\omega _s}$, $2{\omega _c} \cong {\omega _s}$ within the initial values $x(0) = 0, \dot x(0) = 0, \dot v(0) = 0$and$v(0) = 0.5$. The amplitudes of the system ($x$) and NSC ($v$) be about 285.6 and 1,394.2% of the excitation amplitude$\, f$, respectively. The saturation appears for the main system and the controller at (t = 700 sec) and (t = 600 sec), correspondingly. It is importance to note that from Figures 2 and 3 the steady state amplitude of the system with NSC condensed to about 65.55 % from its value without NSC. This wealth that the effectiveness of the NSC (${E_a}$ = the amplitude of the system without controller / the amplitude of the system with controller) regarding 290.27 for the main system ($x$).

(${\eta _s} = 0.02;{\omega _s} = 1;\alpha = 0.894;\beta = 0.0001;\gamma = 0.0001;f = 0.05;\Omega \cong {\omega _s}$).

4.1.   The effects of various parameters

In this branch, we will confirm all effect happening in the parameters of the model with NSC near the measured resonance case. The frequency response equations (FRE) specified by Eqs (3.5) and (3.6) solved and plotted by matching value of parameters which showing in Figures 2 and 3. Figure 4 shows the amplitudes for the system and NSC $a$ and $b$ against detuning value ${\sigma _1}$ as surface plot. In Figure 5(a), we presented the frequency response curves in two-dimensional to explain the behavior of the main system ($a$) before and after NSC ($b$) at different values of ${\sigma _1}$ on case ($a \ne 0, \, \, \, b \ne 0$), moreover, the solid line locate for stable solution and the dashed line for unstable solution. Figure 5(b) indicates bending to the left, it shows obvious soft spring characteristics dominates nonlinearity. Figure 5 shows the frequency–response curves for closed loop case (controller in action) at chosen values to remove the overload risk on the controller for some negative values of detuning parameter ${\sigma _1}$. The figure shows that, the controller work effectively to suppress the main system vibrations at specific frequency bandwidth around ${\sigma _1}$ = 0:0, otherwise the controller will be deactivated automatically.

4.1.1.   Effect of excitation amplitude f

Figure 6, explain that the amplitudes of the system and NSC are monotonic increasing functions of the force amplitude f. The non-zero solutions at ${\sigma _1} = 0$ for the main system are the same when the force is increasing as shown in Figure 6(a–c).

4.1.2.   Effects of linear damping coefficient ${\eta _s}$

On behalf of growing or lessening values for linear damping coefficient ${\eta _s}$ the amplitudes of the system and NSC are trivial as seen in Figure 7.

4.1.3.   Produce of linear damping coefficient ${\eta _c}$ and natural frequency ${\omega _s}$

From Figures 8 and 9, it is seen that the amplitudes of the system and NSC are monotonic lessening functions in linear damping coefficient ${\eta _c}$and natural frequency ${\omega _s}$. But at ${\sigma _1} = 0$, the amplitude of the system is increasing as exposed in Figures 8(a) and 9(a). Furthermore, the steady state amplitude of NSC is shifting to the left when the natural frequency ${\omega _s}$ increased as shown in Figure 9(b).

4.1.4.   The effects of nonlinear parameters $\alpha, \beta$ and $\gamma$

It can expected from Figures 10–12 that, as the nonlinear parameters $\alpha, \beta$ and $\gamma$ increased, the amplitudes of both system and NSC decreased. Additionally, the frequency bandwidth around ${\sigma _1} = 0$ unchanged for the main system as shown in Figures 10(a–c), 11(a–c) and 12(a–c).

4.1.5.   Effect of control signal gain ${\gamma _1}$

Figure 13(a) appears that the amplitude of the system saturated like control signal gain ${\gamma _1}$ increased. Moreover, the amplitude of NSC is a monotonic decreasing function of control signal gain as shown in Figure 13(b).

4.1.6.   Effect of feedback control gain ${\gamma _2}$

Figure 14 illustrates that for the large value of the feedback signal gain${\gamma _2}$, the amplitudes of system and NSC increased. Furthermore, bandwidth for both steady state amplitude is wider.

4.1.7.   Effect of detuning parameter ${\sigma _2}$

Figure 15 shows that once ${\sigma _2}$ increased, amplitudes of both head system and NSC decreased and shifted to the left. As of the frequency response curves, we obtained so as to the lowest amplitude of the system occurs at${\sigma _2} = 0.05, {\sigma _2} = 0$and ${\sigma _2} = - 0.05$ when ${\sigma _1} = - 0.05, {\sigma _1} = 0$and${\sigma _1} = 0.05$, respectively. This means that${\sigma _1} = - {\sigma _2}$.

5.   Verification of analytical solutions using numerical simulation solutions

The system which set by Eqs (2.2) and (2.3) solved numerically at the simultaneous primary in the presence 1:2 internal resonance case where ($\Omega \cong {\omega _s},$ ${\omega _s} \cong 2{\omega _c}$) compared with the analytical solution of the modulating Eqs (2.19) to (2.22) as presented in Figure 16. On the other hand, the continuous lines symbolize the time histories which obtained numerically (using RKM) for Eqs (2.2) and (2.3), and the dashed lines confirm inflection of amplitudes for the coordinates $x$ and $\nu$ at similar values of parameters which using in Figure 3.

6.   Comparison with previously published work

Reference ^[16] studied a harmonically excited dynamical system with a new NIPPF, IRC and PPF controllers. They studied the system with NIPPF analytically by using perturbation performance at primary and 1:1 internal resonance case. They can reduce the vibration amplitude, with zero initial conditions, by using NIPPF, PPF and IRC approximately to 84.87, 82.24, and 46.71%, respectively from its value of the uncontrolled system.

The subject in this article presents the nonlinear system with external excitation force to decrease vibrations by applying the nonlinear saturation controller (NSC). Perturbation technique useful to conclude the frequency response equations at primary and 1:2 internal resonance cases. We succeed to trim down the amplitude of the system, with nonzero initial conditions, approximately to 65.55% forms its resonance value of the non-controlled system. From the study of the effect of parameters and stability regions, we obtained that we can reduce the amplitude by decreasing ${\eta _c}, {\omega _s}, \alpha, \beta$ and$\gamma$. The analysis demonstrates with the purpose of all calculation from the methodical solution is in excellent agreement with the numerical simulation.

7.   Conclusions

A nonlinear saturation controller is proposed to improve the control performance of a dynamical cantilever beam system. The suggested nonlinear controller NSC has been studied for the main system with external excitation force near the considered simultaneous resonance case. According to the introduced control law, the system dynamical model is investigated and then analyzed utilizing perturbation techniques. Stability and the effects of some key parameters of the system are also examined numerically. The regions of stable and unstable of the controlled system are determined and recorded. The influence of the system dynamics with and without controller were explored through different response curves showing the regions of stable, unstable, saddle-node bifurcation, pitchfork bifurcation, and Hopf bifurcation. Finally, numerical confirmations for all obtained analytical results are introduced.

According to the above discussion, the following conclusions can be listed:

1). The measured resonance case is $\Omega \cong {\omega _s}$ and${\omega _s} \cong 2{\omega _c}$ as the worst one.

2). The proposed control NSC method is main process for reducing the vibration amplitudes of the system dramatically.

3). The amplitude for the system reduced near 65.55 % as of its value without NSC. The saturation occurs for the main system and NSC at (t = 700 sec) and (t = 600 sec) correspondingly.

4). The amplitude of the system is monotonic increasing functions headed for the coefficients$f$, ${\gamma _1}$ and monotonic decreasing functions toward the parameters ${\eta _c}, {\omega _s}, \alpha, \beta$ and$\gamma$.

5). Using the saturation controller can avoid the phenomenon of double peaks on the frequency-response curve of the controlled system. Also, NSC can suppress the transient vibrations, thus shorten the time to reach a steady state and can also enhance the efficiency of the vibration reduction and suppress the nonlinear behavior of the system.

6). Both the steady state approximate solutions and the stability analysis based on the integral equation method are in excellent agreement with numerical simulations.

7). The response curves of the system before and after adding the saturation controller appeared various bifurcation phenomena.

Acknowledgments

We would like to thank the reviewers for their positive suggestions, which helped to improve the paper enormously.

Conflict of interest

The authors declare that there is no conflict of interest.

Appendix