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Invariant curves for a delay differential equation with a piecewise constant argument
- Jinghua Liu^{1, 2}Email author
https://doi.org/10.1186/s13662-014-0349-7
© Liu; licensee Springer 2015
Received: 12 September 2014
Accepted: 25 December 2014
Published: 30 January 2015
Abstract
In order to understand the dynamics of a second order delay differential equation with a piecewise constant argument, we investigate invariant curves of the derived planar mapping from the equation. All invariant curves are given in this paper.
Keywords
- difference equation
- invariant curve
- piecewise construction
- characteristic root
- dual equation
1 Introduction
2 Main results
- (C1)
\(0< r_{1}<1<r_{2}\), if and only if \(-2< a<0\).
- (C2)
\(r_{1}=r_{2}=1\), if and only if \(a=0\).
- (C3)
\(r_{1}<0<r_{2}\neq1\) and \(r_{1}\ne-r_{2}\), if and only if \(a<-2\).
- (C4)
\(r_{1}=r_{2}<0 \), if and only if \(a=16\).
- (C5)
\(r_{1}< r_{2}<-1\), if and only if \(a>16\).
Note that the case \(r_{2}>r_{1}>1\) is not listed because the case \(r_{2}>r_{1}>1\) implies \(\frac{(4-a)-(a^{2}-16a)^{\frac{1}{2}}}{4}>1\), i.e., \(-a>(a^{2}-16a)^{\frac{1}{2}}\), which does not hold, and that the case \(0< r_{1}<r_{2}<1\) is not listed because \(0< r_{1}<r_{2}<1\) implies \(0<\frac{(4-a)+(a^{2}-16a)^{\frac{1}{2}}}{4}<1\), i.e., \(0< a<4\), which contradicts the requirement that \(\Delta=a^{2}-16a\geq0\), and that the case \(0< a<16\) is not listed because in this case (4) with \(b=0\) has no continuous solutions, neither \(r_{1}\) nor \(r_{2}\) is real, by [14]. Since we consider \(a\notin\{-2, 4\}\), none of the case \(r_{1}=0\), the case \(r_{2}=0\), and the case \(r_{1}=-r_{2}\ne0\) is listed. Corresponding to the above list, we have the following results.
Theorem 2.1
Proof
The proof is a simple application of well-known results in [14]. The result (i) is given by Theorem 2 of [14], where the characteristic roots \(r_{1}\), \(r_{2}\) satisfy \(r_{2}>1>r_{1}>0\) as shown in (C1). We can deduce the result (ii) from Theorem 8 of [14], where \(r_{1}=r_{2}=1\) as shown in (C2). The proof is completed. □
Theorem 2.2
Proof
Remark
In what follows, we consider the case that either \(a=-2\) or \(a=4\), which is not generic.
For \(a=-2\), (4) is of the form \(f^{2}(x)-3f(x)=-b\), from which we get with the replacement \(y=f(x)\): \(f(x)=3x-b\).
Declarations
Acknowledgements
The author would like to thank the referees for their valuable comments and suggestions, which have helped to improve the quality of this paper. This work was supported by the general item (L0801) of Zhanjiang Normal University.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Cooke, KL, Wiener, J: Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl. 99, 265-297 (1984) View ArticleMATHMathSciNetGoogle Scholar
- Shah, SM, Wiener, J: Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci. 6, 671-703 (1983) View ArticleMATHMathSciNetGoogle Scholar
- Busenberge, S, Cooke, KL: Nonlinearity and Functional Analysis. Academic Press, New York (1982) Google Scholar
- Aftabizadeh, AR, Wiener, J, Xu, J: Oscillatory and periodic properties of delay differential equations with piecewise constant arguments. Proc. Am. Math. Soc. 99, 673-679 (1987) View ArticleMATHMathSciNetGoogle Scholar
- Gyori, I, Ladas, G: Linearized oscillations for equations with piecewise constant arguments. Differ. Integral Equ. 2, 123-131 (1989) MathSciNetGoogle Scholar
- Wiener, J, Cooke, KL: Oscillations in systems of two differential equations with piecewise constant arguments. J. Math. Anal. Appl. 137, 221-239 (1989) View ArticleMATHMathSciNetGoogle Scholar
- Anosov, DV: Geodesic Flows on Closed Riemannian Manifolds of a Negative Curvature. Trudy Mat. Inst. Im. V.A. Steklova, vol. 90 (1967) (Russian) Google Scholar
- Kuczma, M, Choczewski, B, Ger, R: Iterative Functional Equations. Encyclopedia of Mathematics and Its Applications, vol. 32. Cambridge University Press, Cambridge (1990) View ArticleMATHGoogle Scholar
- Matkowski, J: Mean-type mappings and invariant curves. J. Math. Anal. Appl. 384, 431-438 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Nitecki, Z: Differentiable Dynamics: An Introduction to the Orbit Structure of Diffeomorphisms. MIT Press, London (1971) MATHGoogle Scholar
- Sternberg, S: On the behaviour of invariant curves near a hyperbolic point of a surface transformation. Am. J. Math. 77, 526-534 (1955) View ArticleMATHMathSciNetGoogle Scholar
- Ng, CT, Zhang, W: Invariant curves for planar mappings. J. Differ. Equ. Appl. 3, 147-168 (1997) View ArticleMATHMathSciNetGoogle Scholar
- Yang, D, Zhang, W, Xu, B: Notes on a functional equation related to invariant curves. J. Differ. Equ. Appl. 9, 247-255 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Matkowski, J, Zhang, W: Method of characteristic for function equations in polynomial form. Acta Math. Sin. 13, 421-432 (1997) View ArticleMATHMathSciNetGoogle Scholar
- Yang, D, Zhang, W: Characteristic solutions of polynomial-like iterative equations. Aequ. Math. 67, 80-105 (2004) View ArticleMATHGoogle Scholar
- Zhang, WM, Zhang, W: On continuous solutions of n-th order polynomial-like iterative equations. Publ. Math. (Debr.) 76, 117-134 (2010) MATHGoogle Scholar