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:
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
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
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:
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. . 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
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
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.
for all t and x and is bounded.
For the domain t ∈ [0, T], |x| < ∞, |y| < ∞ and x2 + y2 ≥ k2, and by considering the function 𝕍 using Eq.11, we have
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
- 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,
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 + ey3 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 ,
for all t and 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 x2 + y2 ≥ k2, and by considering the following two function 𝕍1 and 𝕍2 as follow
Thus, there exist in the interior of two domains D1 and D2 such that
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.
Then, the governing equation reads
If pψ = pφ = po, then the governing equation reads
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 , 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(ωn) 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:
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
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
3.3 Stability analysis of gyro relative equilibria
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
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
Under the condition of
When we take the assumed values of ωn = −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
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.
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.
A. C. Lazer and P. J. McKenna, On the existence of stable periodic solutions of differential equations of duffing type. Proc. Amer. Math. Soc., 110:125–133, 1990.
B. Mehri, Periodic solutions of a second order nonlinear differential equation. Bull. Austral. Math. Soc., 40:357–361, 1989.
C. Chicone, Ordinary differential equations with applications. Springer, 2000. ISBN-10: 0-387-30769-9.
D.R. Merkin, Introduction to the theory of stability. Springer-Verlag, 1997.
D.W. Jordan and P. Smith, Nonlinear ordinary differential equations: an introduction for scientists and engineers. Oxford University Press, 2007.
E. Leimanis, The general problem of the motion of coupled rigid bodies about a fixed point. Springer Verlag, Berlin, 1965.
E.A. Coddington and N. Levinson, Theory of ordinary differential equations. McGraw Hill, New York, 1955.
F. Wang and H. Zhu, Existence, uniqueness and stability of periodic solutions of a dufffing equation under periodic and anti-peroidic eigenvalues conditions. Taiwanese J. of Mathematics, 19(5):1457–1468, 2015.
G. Morosanu and C. Vladimirescu, Stability for damped nonlinear oscillator. Nonlinear Analysis, 60:303–310, 2005.
H. Chen and Y. Li, Rate of decay of stable periodic solutions of Duffing equations. J. Differential Equations, 236:493–503, 2007.
H. Chen, Y. Li and X. Hou, Exact multiplicity for periodic solutions of duffing type. Nonlinear Analysis: Theory, methods and applications, 55(1:2):115–124, 2003.
H.K. Wilson, Ordinary differential equations. Edwardsvill, III, 1970.
J.B. Scarborough. The gyroscope: theory and applications, Interscience Publishers, Inc., New York, 1958.
J.G. Alaba and B.S. Ogundare, On stability and boundedness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equation. Kragujevac J. of Mathematics, 39(2):255–266, 2015.
L. Cesari, Functional analysis and periodic solution of nonlinear differential equation. Contribution to Differential Equations, 1:149–187, 1963.
M. El-Borhamy, Perturbed rotational motion of a rigid body. M.Sc. thesis, Faculty of Engineering, University of Tanta, Egypt, 2005.
M. El-Borhamy, On the existence of new integrable cases for Euler-Poisson equations in Newtonian fields. Alex. Eng. Journal, 58:733–744, 2019.
M. Lara, Complex variables approach to the short axis mode rotation of a rigid body. Appl. Math. Nonl. Sci., 3(2):537–552, 2018.
M.N. Armenise, C. Ciminelli, F. Dell’Olio and V.M.N. Passaro, Advances in gyroscope technologies. Springer, 2010. ISBN 978–3–642–15493–5.
A.H. Nayfeh, Introduction to perturbation techniques. John Wiley & Sons, 2011.
O.G. Mustafa and Y.V. Rogovchenko, Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations. Nonlinear Analysis, 51:339–368, 2002.
R. Ortega, Stability and index of periodic solutions of an equation of duffing type. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 3:533–546, 1989.
R. Ortega, Topological degree and stability of periodic solutions for certain differential equations. J. London Math. Soc., 42:505–516, 1990.
R. Ortega, Periodic solutions of a newtonian equation: stability by third approximation. Journal of differential equations, 128:491–518, 1996.
R. Reissig, On the existence of periodic solutions of certain non-autonomous differential equation. Ann. Mat. Pura Appl., 85:235–240, 1970.
R. Seydel, New methods for calculating the stability of periodic solutions. Comput. Math. Applic., 14(7):505–510, 1987.
T. Yoshizawa, Stability theory and the existence of periodic solutions and Almost Periodic solutions. Springer-Verlag, 1975. ISBN–I3: 978–0–387–90112–1.
F.E. Udwadia and B. Han, Synchronization of multiple chaotic gyroscopes using the fundamental equation of mechanics. Journal of Applied Mechanics, 75(2):021011:1–10, 2008.
Z. Diab and A. Makhlouf, Asymptotic stability of periodic solutions for differential equations. Advances in Dynamical Systems and Applications, 10(1):1–14, 2016.