Open Journal of Applied Sciences
Vol.08 No.10(2018), Article ID:87909,6 pages
10.4236/ojapps.2018.810036
A Study of Periodic Solution of a Duffing’s Equation Using Implicit Function Theorem
E. O. Eze1, J. N. Ezeora2, U. E. Obasi1
1Department of Mathematics, Michael Okpara University of Agriculture, Umuahia, Nigeria
2Department of Industrial Mathematics and Statistics, Ebonyi State University, Abakaliki, Nigeria
Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: April 23, 2016; Accepted: October 19, 2018; Published: October 22, 2018
ABSTRACT
In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Under appropriate conditions around the origin, a unique periodic solution was obtained.
Keywords:
Implicit Function Theorem, Periodic Solution, Duffing Equation, Banach Spaces
1. Introduction
The well-known implicit function theorem has been employed by many authors to study existence of solution to non-linear differential equations of various types. [1] [2] [3] investigated the existence of solution to ordinary differential equations using implicit function theorem. Other researchers [4] - [10] used implicit function theorem to show the existence of periodic solution for non-linear partial differential equations. The Duffing equation (oscillator):
(1.1)
where a, b, c are real constants and h(t) is continuous, has been widely used in physics, economics, engineering, and many other physical phenomena. Given its characteristic of oscillation and chaotic nature, many scientists are inspired by this nonlinear differential equation given its nature to replicate similar dynamics in our natural world. This equation together with Van der Pol’s equation has become one of the most common examples of nonlinear oscillation in textbooks and research articles. See for instance [11] [12] [13] [14] and the references therein. Due to the importance of the Duffing equation in real world problems, the study of existence of solution of the equation has continued to attract the attention of many researchers. [15] [16] [17] [18] have proposed independently, the existence of periodic solution of Duffing equation of the general form:
(1.2)
where is continuous and 2π-periodic in and .
Motivated by the above results, the purpose of this paper is to study the existence and uniqueness of periodic solution of Duffing equation of the form:
(1.3)
using implicit function theorem where a, b, c are real constants and is continuous with boundary conditions
2. Preliminaries
Definition 2.1. Consider the general non-linear differential equation of the form
(2.1)
where is continuous. The function f in Equation (2.1) is said to be T periodic if for every and some . and for all .
Definition 2.2. A solution x of Equation (2.1) defined on R such that for all is called T periodic solution or T periodic solution.
Definition 2.3. Let E, F be Banach spaces, U an open subset of E and let . Let be a mapping of U into F. f is said to be Frechet differentiable at x0 if there exists a continuous linear mapping; such that
Definition 2.4. Let E, F, G be Banach spaces, let be open set and be a mapping of U into G, with . f is said to be Frechet differentiable with respect to the first variable x at if the following conditions hold.
1) There exists a continuous linear mapping such that such that then , and such that , it follows that.
2) .
3) The mapping is continuous at .
Proposition 2.5. Then condition (1) of the definition 2.4 is satisfied if the partial Frechet derivative exists for in a neighbourhood of and if the mapping is continuous at .
Proposition 2.6. If f is Frechet differentiable with respect to the first variable at , it is Frechet differential with respect to this variable at with the same L1. Moreover, this is unique. L1 is called the strong partial Frechet derivative with respect to the first variable at and denoted by .
Lemma 2.7. (The Banach fixed point theorem) Let E be a Banach space and be a contraction mapping, then f has a unique fixed point in E, i.e. there exists a unique such that .
Lemma 2.8. (The implicit function theorem) Let E, F, G be Banach spaces and let . Set . For arbitrary , let be a mapping satisfying the following conditions.
1) .
2) f is Frechet differentiable with respect to the first variable at .
3) is a linear homeomorphism.
Then there exists a neighborhood of and a unique mapping such that for each the equation has in V1, the unique solution : Moreover, is continuous at y0.
Lemma 2.9. If X and Y are Banach spaces and with and N(A) = {0}, then where N(A) is the Null spaces of A and R(A) is the range space of A. B(X, Y) is the space of bounded linear transformations from X to Y.
3. Main Result
We present in this section, the main result of this paper.
Theorem 3.1. Let = { : x is a class of C2} and equipped with the usual uniform norm C = { : x is continous} with the usual norm, , .
Then, Equation (1.2) is equivalent to
(3.1)
where is defined by
(3.2)
Proof: We first remark that with the norm defined above, is a Banach space. The strategy for the proof involves application of the implicit function theorem to the function f defined in Equation (3.1). We split the proof into steps.
Step 1: . This follows trivially from the definition of f: hence
(3.3)
Step 2: f is Frechet differentiable with respect to x at (0; 0). Observe that
Consequently,
(3.4)
Combining (3.3) and (3.4), we obtain that f is Frechet differentiable with respect to the first variable at (0, 0).
Step 3: defined by is a linear homeomorphism.
The mapping is linear and continuous and hence bounded. It is also an onto mapping. Linear homeomorphism would have been established if the mapping is shown to be one to one. This is equivalent to requiring that
(3.5)
with
(3.6)
be non-critical.
It suffices to place appropriate conditions on the constants a, c such that Equation (3.5) is solvable. The auxiliary equation of (3.5) is .
Case I:
If , and where k is a natural number, then and
(3.7)
for arbitrary constants and . Clearly
(3.8)
and the solution is non-trivial.
Case II:
If and , then condition
(3.9)
is satisfied only by the trivial solution z = 0.
Case III:
If and , only the trivial solution exists. Most generally, put
(3.10)
for some real numbers u, v.
1) Choose c and a such .
2) Choose c and a such that then is non-critical [19] .
3) Choose c and a such that is non-singular where is fundamental matrix of Equation (3.5) with the identity matrix [20] .
Thus with any of these conditions imposed, one deduces the one to oneness of . Hence by Lemma 2.8 exists as a bounded linear operator. Linear homeomorphism of follows. Existence of a unique solution is now assured by the implicit function theorem.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Eze, E.O., Ezeora, J.N. and Obasi, U.E. (2018) A Study of Periodic Solution of a Duffing’s Equation Using Implicit Function Theorem. Open Journal of Applied Sciences, 8, 459-464. https://doi.org/10.4236/ojapps.2018.810036
References
- 1. Chow, S.N. and Lasota, A. (1972) An Implicit Function Theorem for Non-Differentiable Mappings. Proceedings of the American Mathematical Society, 34, 141-146. https://doi.org/10.1090/S0002-9939-1972-0291527-7
- 2. Chicone, C. (1996) Ordinary Differential Equation with Applications. Springer-Verlag, New York.
- 3. Hartman, P. (2014) Ordinary Differential Equation. 2nd Edition, Willey and Sons Publications, New York, 235-239.
- 4. Kreici, P. (1984) Hard Implicit Function Theorem and Small Periodic Solutions to Partial Differential Equations. Commentationes Mathematicae Universitatis Carolinae, 25, 519-536.
- 5. Magnus, R. (1974) The Implicit Function Theorem and Multi Bump Solutions of periodic Partial Differential Equation. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 136, 559-583.
- 6. Mishra, L.N. and Sen, M. (2016) On the Concept of Existence and Local Attractivity of Solutions for Some Quadratic Volterra Integral Equation of Fractional Order. Applied Mathematics and Computation, 285, 174-183.
- 7. Mishra, L.N., Agarwal, R.P. and Sen, M. (2016) Solvability and Asymptotic Behavior for Some Nonlinear Quadratic Integral Equation Involving Fractional Integrals on the Unbounded Interval. Progress in Fractional Differentiation and Applications, 2, 153-168.
- 8. Deepmala (2014) A Study on Fixed Point Theorem for Nonlinear Contraction and Its Applications. Ph.D. Thesis, Pt. Ravishankar Shukla University, Raipur, India.
- 9. Mishra, V.N. (2007) Some Problems on Approximations of Functions in Banach Spaces. Ph.D. Thesis, Indian Institute of Technology, Roorkee, India.
- 10. Deepmala and Pathak, H.K. (2013) A Study on Some Problems on Existence of Solutions for Nonlinear Functional-Integral Equation. Acta Mathematica Scientia, 33, 1305-1313.
- 11. Puu, T. (2000) Attractors, Bifurations & Chaos: Nonlinear Phenomena in Economics. Spring-Verlag, Berlin, Heidelberg.
- 12. Ueda, Y. (1979) Randomly Transitional Phenomena in the System Governed by Duffing’s Equation. Journal of Statistical Physics, 20, 181-196. https://doi.org/10.1007/BF01011512
- 13. Chen, H.B. and Li, Y. (2000) Existence, Uniqueness and Stability of Periodic Solution of an Equation of Duffing Type. AIMS Journal, 10, 10-20.
- 14. Zhang, W.B. (2005) Differential Equations, Bifurcations, and Chaos in Economics. World Scientific, Singapore. https://doi.org/10.1142/5827
- 15. Njoku, F.I. and Omari, P. (2003) Stability Properties of Periodic Solutions of a Duffing’s Equation in the Presence of a Lower and Upper Solutions. Applied Mathematics and Computation, 135, 471-490. https://doi.org/10.1016/S0096-3003(02)00062-0
- 16. Pedro, J.T. (2004) Existence and Stability of Periodic Solutions of a Duffing’s Equation by Using a New Maximum Principle. Mediterranean Journal of Mathematics, 1, 470-486.
- 17. Sani, G. and Alain, H.N. (1989) N-Cyclic Function and Multiple Subharmonic Solutions of Duffing’s Equation.
- 18. Yuji, L. and Weiguo, G. (2004) Positive Solution of Non-Linear Duffing’s Equation with Delay and Variable Coefficients. Tamsu Oxford Journal of Mathematical Sciences, 20, 235-255.
- 19. Hale, J.K. (1963) Oscillation in Non-Linear System. McGraw Hill, New York.
- 20. Hale, J. and Taboas, P.Z. (1978) Interaction of Forced Damping in a Second Order Evolution Equation.