Abstract
The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly by obtaining the period and the solution. The period is given in terms of the complete elliptic integral of the first kind and the solution involves Jacobian elliptic functions. We solve the cubic-quintic Duffing equation under arbitrary initial conditions. Physical applications are provided. The solution to the mixed parity Duffing oscillator is also formally derived. We illustrate the obtained results with concrete examples. We give high accurate trigonometric approximations to the Jacobian function cn.
1. Introduction
It is well known that many engineering problems are not linear and their analytical solutions are not easy to obtain. Disturbance methods are among the known methods for solving nonlinear problems, which are based on the existence of small/large parameters, the so-called disturbance parameters. Our approach is different from known solutions to this problem [1–5]. On the contrary, here, we express the solutions without imaginary quantities: both frequency and modulus are real numbers. The quintic term appearing in the cubic-quintic Duffing equation makes this nonlinear oscillator not only more complex but also more interesting to study.
2. The Analytical Solution to the Cubic-Quintic Duffing Equation
Let , , , , and be arbitrary real numbers. We will solve the initial value problem:
We will assume that . Multiplying (1) by and integrating it with respect to gives
Letwhere the parameter values , , , , , and are to be determined. If the solution in (3) is periodic, it will have the same period as the function cn and this period may be evaluated by means of the formula
In the case, when , we may approximate the value of using the formula
The error for this approximation is given by
On the contrary, we may obtain approximate trigonometric solution making use of the following approximation formula:
See Table 1, for the error = .
A more accurate trigonometric approximation may be obtained using the formulabeing
See Table 2, for the error = .
Let
Introduce the notation:
Definition 1. The discriminant to the i.v.p. (1) is defined as
2.1. First Case:
We define Inserting the ansatz (3) into (10), we obtain
Equating to zero the coefficients of in the numerator of the last expression gives an algebraic system. Solving it, we obtain
For these choices, we obtain the following system:
Eliminating from system (15) gives the sexticwhere
Sextic (16) is solvable by radicals. Indeed, let
Then,where
The discriminant to cubic (19) is
Cubic (19) has three real roots and at least one of them must be positive. Indeed, let , , and be the roots to this cubic. Then,
Since , at least one of the numbers , , and must be positive. We choose the closest to positive root to cubic (19) so that, in view of (17), the number will be the closest to zero real root to the sextic in (15). Observe also that the condition implies that . Thus, if , then . Moreover, the discriminant to the cubic (17) and have the same sign.
The numbers and are determined from the initial conditions:
The number is a solution to the sextic:
Example 1. LetSextic (15) readsThe roots to this sextic areWe choose the value . The values of and are determined from the initial conditions. They readThe exact solution to the i.v.p.,readsThe solution is periodic with period . The approximate trigonometric solution is given byThe error of this trigonometric approximant compared with the exact solution on equals 0.00352816, see Figure 1.
2.2. Second Case:
Let . Inserting the ansatz (3) into (10), we obtain
Equating to zero the coefficients of gives an algebraic system. Solving it gives
The system reduces to
Eliminating from this system gives the sexticwhere
Sextic (34) has at least one real root. Indeed, let () be its roots. Then,
We will choose the closest to zero real root to sextic (34). The values for and are determined from the initial conditions:
The number is found from the algebraic equation:
Example 2. Let . The i.v.p. problem to be solved isThis problem has a negative discriminant . Sextic (35) readsThe roots areWe choose . The values for and areThe exact solution is given bySee Figure 2, for a comparison with the numerical solution.
2.3. Third Case:
In this case, we have two roots to the cubic:
At least one of these two numbers must be positive. So, we proceed the same way as we did for a positive discriminant.
2.4. Two Particular Cases
2.4.1. First Particular Case:
Let
The discriminant to the i.v.p. (46) equals
If , then the exact solution is given bywhere
Assume now a negative discriminant. The exact solution reads
We claim that . Indeed, let
Then, . However,
From the last identity, it is clear that .
Assume now that . In this case, we have the following solutions:
Solutions (53) are called solitons. They arise in soliton theory.
2.4.2. Second Particular Case:
Let
The solution has the form
The values of and are obtained by solving the two sextics:whereand
3. Applications
The cubic-quintic Duffing oscillator has many interesting applications in soliton theory, optics, nonlinear circuits, plasma physics, and other areas of science and engineering.
3.1. Nonlinear Odd Parity Oscillators
Let us consider the nonlinear oscillator:where is and odd function:
We may approximate the function by means of a Chebyshev polynomial on some interval () as follows:where
Then, we may obtain an approximated analytic solution to oscillator (59) by solving the cubic-quintic Duffing oscillator:
Example 3. Let us obtain an approximate analytical solution to the pendulum equation:having the analytical solution [9]The Chebyshev approximant of on readsWe may rationalize this expression up to to obtainThe square mean error isThus, our aim is to solve the cubic-quintic Duffing oscillator:Let and . For these data, we have a positive discriminant and the solution to the initial value problem,may be obtained making use of formula (48).
3.2. The Duffing–Helmholtz Oscillator
Let us consider the i.v.p.:
Suppose that the function is the solution to the cubic-quintic Duffing equation:
Then, the function,is the solution to the i.v.p. (71) provided that is a solution to the quartic:
3.3. Nonlinear Conservative Oscillators
Suppose we are given to sole the i.v.p.:
Assume that . We approximate the function by means of a cubic polynomial on using Chebyshev approach so thatwhere
We now replace the i.v.p. (75) with the i.v.p.:
Let
The problem reduces to the i.v.p.:
This is a Duffing–Helmholtz oscillator (71) with
4. Analysis and Discussion
We have solved the cubic-quintic Duffing oscillator equation for any given arbitrary initial conditions. In [1], authors considered the particular case:under the restrictions,
These conditions, however, are too restrictive. In [6], author considered the ansatz:
This approach does not allow to determine the sign of . On the contrary, this ansatz sometimes gives complex values for or , which makes it difficult to interpret the obtained solution physically. Our approach avoids obtaining such complex values.
Other authors solved this equation using perturbative methods [2, 4, 7–9]. In [5], author studied the stability analysis to a cubic-quintic Duffing equation. There are other numerical and analytical methods that allow to solve this oscillator equation.
The cubic-quintic Duffing oscillator may also be solved making use of perturbative methods. One of them is the famous Krylov–Bogoliubov–Mitropolsky method (KBM). For example, let us consider the following oscillator:
Using the Krylov–Bogoliubov–Mitropolsky method gives the following approximate analytical solution:where
For the damped oscillator,
The KBM giveswhere
The constants and are determined from the initial conditions. These results are also valid for the cubic Duffing oscillator :
5. Conclusions
The cubic-quintic Duffing oscillator has been solved exactly for arbitrary initial conditions. The obtained results may be applied to solve strongly nonlinear conservative oscillators like the pendulum oscillator equation. We may go further by considering a damped oscillator of the form . In the case when , we may approximate the function by means of some cubic-quintic polynomial using Chebyshev approximation formulas. The solution is then assumed in the form , where is some parameter having a value near and is the exact solution to some cubic-quintic Duffing oscillator equation.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.