## 1 Introduction

Many applied problems represented by dynamical systems have many difficulties to obtain their solutions in explicit forms definitely when these problems are modelled by (non-autonomous)nonlinear systems, cf. [5,7,12]. Therefore, the directions of progress to analyze qualitatively for obtaining the main mathematical properties now become must. Most of these properties are stuck in the presence of periodic solutions, the bifurcations and the route to the chaotic behaviours or more less based on their dynamics as well. One of these celebrated applied problems is the general planar motion of a particle exerted by conservative and nonconservative fields that can be represented by the following system:

*f*is a nonlinear function satisfying certain conditions specified in the tackled problem.

The nonlinear system represented by Eq.2 is mostly used for mathematical modelling of various engineering phenomena and it is attracting engineers and mathematical researchers due to its rich applications. A great deal of papers and books published during the last decades have extensively studied the qualitative properties of Eq.2 in directions of existence and uniqueness, stability, periodicity(quasi-periodicity), bifurcation, chaos and some numerous particular properties, cf. [2, 3, 15, 21, 25, 27].

In particular, one of the fruitful special cases of Eq.2 tackled in numerous applications for their qualitative constant interests is the following non-autonomous undamped duffing equation

*c*is the coefficient of linear damping and

*r*(

*t*) is the driven function. Chen et al. [11] studied it to obtain the exact multiplicity of periodic solutions with

*f*(

*x,t*) and

*r*(

*t*) as continuous and 2

*π*periodic functions of time. In [29], it is studied the existence, uniqueness and asymptotic stability of periodic solutions of some special type of Van Der Pol oscillators under periodic time excitation is studied. However, so many literatures have been discussed on the qualitative analysis of Eq.2 and its periodic solutions by different techniques, cf. [1, 8, 9, 10, 14, 23, 24, 26].

Motivated by this argument, in this work, we discuss the stability and the existence of periodic solutions as well as the transition to chaos for very specific type of non-autonomous nonlinear ODEs, so-called the Rayleigh–Duffing equation

*h*(

*ẋ*) and

*f*(

*x,t*) are nonlinear continuous functions and

*r*(

*t*) is a continuous periodically driven function. In general, the most general engineering applications of the Rayleigh–Duffing equation (Eq.5) are modelling of RLC circuits with actual voltage, the planar motion of a particle under the exciting and dissipative forces and some of gyro motions under external torques, cf. [19].

Here, the function *h*(*ẋ*) represents the dissipative forces that are considered as a general bounded function or have the generalized form of Rayleigh damping and *f* (*x,t*) represents the conservative and nonconservative forces of duffing type.

Using a dimensionless small parameter, denoted by *ɛ*, the Rayleigh–Duffing equation can be transformed to the following perturbed form:

*ω*represents the natural frequency of the system and 𝔽 represents a perturbed force on the system.

_{n}The concept of chaotic behaviour is important due to its existence profusely in complex systems; therefore, the dynamic application of Eq.5 will concern some what a part of this study. In this work, an interesting application of Eq.5 being the motion of axi-symmetric gyroscope mounted on a vibrating base with periodic excitation is considered, cf. [28]. The general treatment of the governing equation is handled by sides of stability analysis and existence of periodic solutions. Moreover, the theoretical results are applied to the tackled application besides the study of bifurcation and transition to chaos using the perturbed form of the differential equation. Melnikov’s method is used to clarify the range of chaotic behaviour affected by the change in main parameters of the system. Numerical verification by using numerical solution diagrams and phase plane trajectories for proving the deduced theoretical results of the gyro motion are considered.

This work is organized as follows: In section 2, some theoretical results for the Rayleigh–Duffing equation are presented. Section 3 is concerned with a dynamic application using an example of gyro motion governed by the Rayleigh–Duffing equation to verify the theoretical results. In the last section, the conclusion is given.

## 2 Theoretical results

Let us assume that, the Rayleigh–Duffing equation possesses a *T*-periodic solution in the presence of continuous functions *h*(*ẋ*), *f* (*x,t*) and *r*(*t*), with the later two T-periodic with respect to time. In addition to, it is assumed that the following identity at the equilibrium point *x** = 0

*h*(0) +

*f*(

*x,t*) = 0 is

*x*=

*x**. Hence, rewriting Eq.5

From Eq.10, it is easy to obtain the general conditions for energy decaying, but on the other hand one can obtain a general representation of energy function (i.e. Lyapunov’s function) as

*|x| <*∞,

*|y| <*∞, 0 ≤

*t*≤

*T*and

*x*

^{2}+

*y*

^{2}≥

*k*

^{2}where

*k*∈ ℝ

^{+}.

Thus, the following theorem gives general imposed conditions based on Eq.5 to obtain asymptotically stable solutions.

*The Rayleigh-Duffing equation has an asymptotically continuable stable solution if the following conditions are satisfied:*

*i)**h*(*y*)*and f*(*x*,*t*)*are locally Lipschitzian in y and x respectively, and r*(*t*)*is continuous on*ℝ.*ii)**f*(*x,t*)*and r*(*t*)*are periodic in t of period T.**iii)*$F(x,t)={\int}_{0}^{x}f(u,t)\mathrm{du}>-k$ *for all t and x and*$\frac{{F}_{t}}{\sqrt{F(x,t)+k}}$ *is bounded.*

For the domain *t* ∈ [0, *T*], *|x| <* ∞, *|y| <* ∞ and *x*^{2} + *y*^{2} ≥ *k*^{2}, and by considering the function 𝕍 using Eq.11, we have

*i*,

*ii*and

*iii*are satisfied. Then, the conclusion holds.

The existence of periodic solutions of the Rayleigh–Duffing equation can be proved on the basis of Schauder’s fixed-point theorem. General conditions to obtain at least one limit cycle are stated and proved in the following theorem.

*The Rayleigh–Duffing equation has at least one T-periodic solution if the following conditions are satisfied,*

*i)**f*(*x*,*t*)*is locally Lipschitzian in x and r*(*t*)*is a continuous on*ℝ.*ii)**f*(*x*,*t*)*and r*(*t*)*are periodic in t of period T and* .${\int}_{0}^{T}r\left(t\right)\mathrm{dt}=0$ *iii)**| f*(*x*,*t*)| ≤ |*r*(*t*)|*uniformly in t.**iv)**the function h*(*y*)*is bounded, i.e*|*h*(*y*)| <*L.**v)**f*(*x*,*t*)*sgnx*≥ 0.

Let us rewrite Eq.5 in its perturbed form as follows,

*ω*here represents an arbitrary positive constant. Hence, if

_{n}*ɛ*≈ 0 then we obtain a homogenous equation with

*ω*periodic solution but for 0

_{n}*< ɛ <*1 all periodic solutions and their first derivative are uniformly bounded. So that, let

*x*(

*t*) =

*x*(

*t*+

*T*) be a solution of Eq.13 and

_{s}(

*t*+ 0,

*t*) − 𝔾

_{s}(

*t*− 0,

*t*) = 1. This implies that

*q*(

*t*+

*T*) =

*q*(

*t*), then we obtain the following representation of the solution

*x*(

*t*)

*q*(

*t*) to derive the following estimates

*|x| > x*,

_{o}*t*∈ [0,

*T*] it follows that |

*x*(

*t*)| ≥

*x*for 0 ≤

_{o}*t*≤

*T*is excluded, therefore there exits

*τ*such that 0

*< τ < T*then

*x*(

*τ*)

*< x*. Applying the mean value theorem to an arbitrary interval [

_{o}*τ,t*] ⊂ [

*τ,T*], then we have

*R*→ 0, then we have the following priori estimates

These estimates ensure at least the existence of one periodic solution in future, then the conclusion holds.

The following theorem ensures the existence of periodic solutions if the damping function is a Rayleigh damping term *h*(*y*) = *cy* + *ey*^{3} where *c* and *e* are real constants under the following stated conditions.

*The Rayleigh-Duffing equation under the Rayleigh damping term for t* ∈ ℝ^{+}*and x* ∈ [*a,b*] *has at least one periodic solution of period T if the following condition are satisfied,*

*i)**h*(*y*)*and f*(*x,t*)*are locally Lipschitzian in y and x respectively and r*(*t*)*is a continuous on*ℝ.*ii)**f*(*x,t*)*and r*(*t*)*are periodic in t of period T ,**iii)*$F(x,t)={\int}_{0}^{x}f(u,t)\mathrm{du}>-k$ *for all t and*$\frac{{F}_{t}}{\sqrt{F(x,t)+k}}$ *is bounded.**iv)**h*(*y*)*y >*0*for c, e*∈ ℝ.*v)**There exists a,b, a < b, f*(*a,t*) −*r*(*t*) ≥ 0*and f*(*b,t*) −*r*(*t*) ≤ 0.*vi)**| f*(*t,x*)| + |*r*(*t*)| ≤ ℓ.

For the domain *t* ∈ [0,*T* ], *|x| <* ∞, *|y| <* ∞ and *x*^{2} + *y*^{2} ≥ *k*^{2}, and by considering the following two function 𝕍_{1} and 𝕍_{2} as follow

_{1}→ ∞ and 𝕍

_{2}→ ∞ uniformly for (

*x,t*) as

*|y|*→ ∞.

Thus, there exist in the interior of two domains *D*_{1} and *D*_{2} such that

*x*(

*t*) such that |

*x*(

*t*)| + |

*y*(

*t*)| is bounded for all

*t*≥ 0, so that all solutions exist in future and one of them is bounded then there exists at least a periodic solution of period

*T*in future.

## 3 Gyro dynamic application

The Rayleigh–Duffing equation has rich applications in the gyro dynamics, cf. [6, 13, 19]. In our case, the application represents the motion of axi-symmetric gyroscope(Lagrange’s gyroscope) mounted on a vibrating base exerted by a sinusoidal periodic force along the vertical fixed axis(*OZ*) as shown in Fig.1.

The motion can be described by the Routhian’s function(ℜ) as a function of the Euler’s angles *θ* (nutation), *ψ*(precession), and *φ*(spin), cf. [16, 17, 18, 28],

*p*and

_{φ}*p*are the corresponding conserved angular momenta to the cyclic coordinates

_{ψ}*φ*and

*ψ*, respectively.

*m*as the mass of the gyroscope,

*A*,

*B*and

*C*are the principal moments of inertia along the moving axes

*Ox*,

*Oy*and

*Oz*respectively,

*z*is the distance along the polar axis(

_{c}*Oz*) of the centre of gravity(C.G.) of the gyro from its point of support(

*O*) and

*d*(

*t*) =

*d*

_{0}sin

*ωt*is time varying amplitude of the vertical support motion that has constant amplitude

*d*and forced frequency

_{o}*ω*along the vertical fixed axis (

*OZ*).

Then, the governing equation reads

If *p _{ψ}* =

*p*=

_{φ}*p*, then the governing equation reads

_{o}*θ*=

*x*,

The vibrating axi-symmetric gyro equation(Eq.38) can be drawn into the following general form:

*According to the conditions of* theorem 1, *the solution of the vibrating axi-symmetric gyro equation is globally asymptotically stable.*

*According to the conditions of* theorem 3, *the vibrating axi-symmetric gyro equation has at least one periodic solution.*

### 3.1 Approximate form of the periodic solution

In [20], it is found that the straightforward expansion methods fails sometime to obtain a form of periodic solution for the linear problem with periodic coefficient, for instance Mathieu’s equation. Therefore, it is suggested to use the method of strained parameters by expanding the solution(*θ*) and the natural frequency(*ω _{n}*) in powers of

*ɛ*by using the perturbed form of vibrating axi-symmetric gyro equation (Eq.46). Thus, we seek the periodic solution by using the following uniform expansions:

*ɛ*

^{0},

*ɛ*

^{1},

*ɛ*

^{2}, we get the following differential equations,

*a*and

*φ*are arbitrary constants,

### 3.2 Stability of the approximate periodic solution

By rewriting the perturbed form of vibrating axi-symmetric gyro equation (Eq.46) to have the form

*a*and

*φ*are replaced with their average values over the period

*a*and

*φ*are constants in taking the values of average and apply the method of averaging, we obtain

*ψ*=

*ω*+

_{n}t*φ*.

Once the integrals have been evaluated, then we have first order differential equations to obtain the amplitude and the phase angle and an approximate solution is obtained. In the case of existence of limit cycles, then the amplitudes of possible ones can be obtained from

*a*=

*a*

_{1},

*a*

_{2},

*a*

_{3},...), then we obtain the amplitudes of the periodic solutions, consequently to get the condition of stability of the limit cycle, the following condition must be satisfied

*α*= 10,

*β*= 1,

*c*= 0.1,

*e*= 0.05, Γ = 1 and

*ω*= 3, a stable limit cycle is obtained with radius around one, which roughly fits the theoretical result.

### 3.3 Stability analysis of gyro relative equilibria

From Eq.38

*π*), let the disturbance of motion be at (0,0) and using Lyapunov function,

*V >*0

*P*(

*t*) > 1 and

The inequalities in Eq.71 are conditions for the asymptotic stable solution of the vibrating axi-symmetric gyro equation at the fixed point (0,0). We can verify this condition by the following numerical case: *α* = 10, *β* = 1, *c* = 0.5, *e* = 0.2, *ω* = 3 and Γ = 1 as shown in Fig.3 to assure the theoretical conditions at (0,0).

The second relative equilibrium points are at

*V*(

*t,x,y*) > 0 under the condition

*P*(

*t*) − 3

*π*

^{2}

*Q*(

*t*) > 0. If we test the conditions for stability where

*π*, 0).

### 3.4 Homoclinic bifurcation and transition to chaos

To investigate the existence of the homoclinic bifurcation, we use firstly the following approximate or perturbed equation of the gyro system to deduce such critical values using Melnikov’s function,

*Eq.76 admits a homoclinic bifurcation if**or*

Under the condition of

When we take the assumed values of *ω _{n}* = −1, Ω = 1 as a special case to deduce the critical values of the system that generates the homoclinic bifurcation of vibrating axi-symmetric gyro equation, then the following unperturbed equation of the system(

*ɛ*≈ 0) reads

*M*(

*t*) = 0, we have

_{o}*c*= 0.5,

*ω*= 1,

*e*= 0 and

*e*= 0.1 then it is around

*e*= 0 and

*e*= 0.1 respectively.

To obtain a large insight of the chaotic behaviour, we analyze the nonlinear behavior of the vibrating axi-symmetric gyro equation numerically by using the Range–kutta fourth-order method solver to determine the transition to chaotic stages using the phase plane trajectories. In Fig.5 and Fig.6, it can be easily noticed that the output of the system transit from double route bifurcation to complex chaotic behaviour when the normalized amplitude Γ is increased further. By the way, one can also notice that the damping coefficient *e* has an effect to delay or to slow down the transition to the chaos region.

## 4 Conclusion

In this work, special type of nonlinear ordinary differential equations called the Rayleigh–Duffing equation is qualitatively studied by seeking the stability of solutions and existence of periodic solutions via the fixed point method and the second method of Lyapunov. An engineering application governed by the Rayleigh–Duffing equation represented by a motion of vibrating axi-symmetric gyro is investigated based on the deduced theoretical results. In general, it is concluded that the gyro motion under a driven periodic force is affected by the variation of the normalized excitation amplitude from stable or periodic to a complex chaotic dynamic. An approximate measure of the periodic solution and its amplitude using the perturbed forms is deduced. The stability of equilibria of the gyro motion have been explicated using Lyapunov stability analysis. The existence of the homoclinic bifurcation and the transition to chaos by obtaining the range of chaotic behaviour with respect to the normalized value of excitation amplitude are shown. Lastly, all theoretical results are verified and fit well with the numerical ones.

^{}

**Conflict of Interest:** The authors declare that they have no conflicts of interest.

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