Analytical Approximant to a Quadratically Damped Forced Cubic-Quintic Duffing Oscillator

The cubic-quintic Duffing oscillator of a system with strong quadratic damping and forcing is considered. We give elementary approximate analytical solution to this oscillator in terms of exponential and trigonometric functions. We compare the analytical approximant with the Runge–Kutta numerical solution. The approximant allows us to estimate the points at which the solution crosses the horizontal axis.


Introduction
In this paper, some novel analytical and numerical techniques are introduced for analyzing and solving nonlinear ordinary differential equations (NODEs) that are associated to some strongly nonlinear oscillators such as a quadratically damped cubic-quintic Duffing equation.
ere are many numerical and analytical approaches that were applied for solving the second-order nonlinear oscillator equations. For instance, both the homotopy perturbation method (HBM) and MTS technique were applied for analyzing a forced Van der Pol (VdP) generalized oscillator to obtain the amplitudes of the forced harmonic and super and subharmonic oscillatory states [1]. Also, Melnikov's method was employed for analyzing a mVdPD equation and deriving analytical criteria for the appearance of horseshoe chaos in chemical oscillations [2]. He et al. [3] used the Poincare-Lindstedt technique (PLT) for solving and analyzing the hybrid Rayleigh-Van der Pol-Duffing equation. Moreover, the homotopy analysis method (HAM) was employed for analyzing the DVdP oscillator [4]. Both methods of differentiable dynamics and Lie symmetry reduction method were devoted for analyzing the DVdP-type oscillator [5]. e principal feature associated with quadratic damping is a discontinuous jump of the damping force in the equation of motion whenever the velocity vanishes such that the frictional force always opposes the motion.

First Case: Undamped and Unforced Cubic-Quintic Duffing Equation.
Let us consider the i.v.p.
Assume the ansatz where the function v � v(t) is the solution to some Duffing equation e numbers v 0 and _ v 0 are determined from the initial conditions. Observe that From (5), it follows that Observe that We have Eliminating p, q, μ, v 0 , and _ v 0 from the system e values for p, q, and μ are On the other hand, the solution to i.v.p. (5) is given by where is called the discriminant to Duffing equation (5). In the case when Δ > 0 , this solution may also be written in the form where 2 e Scientific World Journal Since Δ < 0, necessarily q < 0. From the equality it is evident that δ > 0. en, the solution to i.v.p. (5) reads where Observe that e exact solution is given by where See Figure 1.

Solution by Means of He's Frequency Method.
Let He's method assumes the solution in the ansatz form e frequency is evaluated by means of the formula e Scientific World Journal en,

Solution by Means of a Simple Trigonometric Ansatz.
As in He's approach, we assume the solution in ansatz form (28) so that We choose the frequency ω so that is last formula looks like He's formula (30). e difference is ω trigo − ω He � (c/16)A 4 . is suggests to consider the following κ-parameter solution: where the number A is a solution to the sextic e number λ is chosen in order to get as small residual error as possible.

Solution by Means of an Improved Trigonometric
Assume the ansatz Let e numbers λ, μ, and ω are found from the conditions e frequency ω is found from the quadratic equation   e Scientific World Journal .
(43) e numbers λ, μ, υ, and ω are found from the conditions e frequency ω is found from the cubic equation
(51) en, where e value of c 0 is found from the initial condition x ′ (0) � _ x 0 , and it is a solution to the sextic e number ρ is a free parameter that is chosen in order to minimize the residual error. In particular, when ε ⟶ 0, we obtain approximate trigonometric solution to the undamped cubic-quintic Duffing equation Example 2. Let us consider the i.v.p.
e approximate analytical solution for ρ � 0.0084 is where the function f(t) is given by (52) with ε � 0.25 , α � 1, β � 5, c � 10, c 0 � − 0.0983, and ρ � 0.0084 (see Figure 2). e obtained results may be applied to solve the pendulum equation with quadratic damping Indeed, we may use the approximation and then we replace i.v.p. (58) with the i.v.p.
Assume the ansatz where the function y � y(τ) is the exact solution to the i.v.p.
e numbers ρ and κ are free parameters that are chosen in order to minimize the residual error Observe that when ε ⟶ 0, ((1 − exp(− εκt))/εκ) ⟶ t and then we obtain the exact solution to the undamped and 6 e Scientific World Journal unforced cubic-quintic oscillator (3). So, we expect accurate approximate analytical solution for small ε. is approach is more accurate, but here the solution involves elliptic functions and the solution is not elementary.

ird Approach.
Let us consider the i.v.p.
Assume the ansatz . (67) Proceeding in a similar way as in the first approach, we may choose Here we have two free parameters ρ and λ that are chosen in order to get as less residual error as possible. e numbers c and d are determined from the initial conditions as follows: e number c is a solution to the octic Assume the ansatz where the function u � u(t) is a solution to some i.v.p.
e suitable constants p, q, r, d 0 , and d 1 are to be determined. Define the residual function − F 0 cos(ωt) − F 1 sin(ωt).