# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Bifurcations of periodic solutions of delay differential equations. (English) Zbl 1027.34081
The author extends the method of {\it J. L. Kaplan} and {\it J. A. Yorke} [J. Differ. Equations 23, 293-314 (1977; Zbl 0307.34070)] to prove the existence of periodic solutions with certain period in scalar delay differential equations of the type $\dot x(t)= F(x(t), x(t-r), x(t-2r))$, where $F$ satisfies the relation $F(x,y,-x)=-F(-x,-y,x)$. For $F$ depending on parameters, the paper gives conditions under which Hopf and saddle-node bifurcations of periodic solutions occur. Moreover, the author provides examples showing that Hopf and saddle-node bifurcations often occur infinitely many times.

##### MSC:
 34K18 Bifurcation theory of functional differential equations 34K13 Periodic solutions of functional differential equations
Full Text:
##### References:
 [1] Chen, Y.: The existence of periodic solutions of the equation x \dot{}$(t)=-f(x(t)$,x(t-${\tau}$)). J. math. Anal. appl. 163, 227-237 (1993) · Zbl 0755.34063 [2] P. Dormayer, A.F. Ivanor, Symmetric periodic solutions of a delay differential equation, in: Dynamical Systems and Differential Equations, Vol. 1 (1998) (added volume to Discrete Continuous Dyn. Systems), Southwest Missouri State University, Springfield, pp. 220--230. · Zbl 1304.34119 [3] Ge, W.: Existence of many and infinitely many periodic solutions for some types of differential delay equations. J. Beijing inst. Technol. 1, 5-14 (1993) · Zbl 0807.34083 [4] Gopalsamy, K.; Li, J.; He, X.: On the construction of kaplan--Yorke type for some differential delay equations. Appl. anal. 59, 65-80 (1995) · Zbl 0845.34073 [5] Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems and bifurcation of vector fields. (1983) · Zbl 0515.34001 [6] Hale, J.: Theory of functional differential equations. (1977) · Zbl 0352.34001 [7] M. Han, On the existence of symmetric periodic solutions of a differential difference equation, Differential Equations and Control Theory, Lecture Notes in Pure and Applied Math. Series, Vol. 176, Marcel Dekker, New York, 1995, pp. 73--77. [8] Jones, G. S.: Periodic motions in Banach space and application to functional differential equations. Contrib. differential equations 3, 75-106 (1964) [9] Kaplan, J.; Yorke, J.: Ordinary differential equations which yield periodic solutions of differential delay equations. J. math. Anal. appl. 48, 317-324 (1974) · Zbl 0293.34102 [10] Kaplan, J.; Yorke, J.: On the nonlinear differential delay equation x \dot{}$(t)=-f(x(t),x(t-1))$. J. differential equations 23, 293-314 (1977) · Zbl 0307.34070 [11] Saupe, D.: Global bifurcation of periodic solutions to some autonomous differential delay equations. Appl. math. Math. comput. 13, 185-211 (1983) · Zbl 0522.34067 [12] Wang, K.: On the existence of nontrivial periodic solutions of differential difference equations. Chinese ann. Math. 11B, 438-444 (1990) · Zbl 0729.34047 [13] Wen, L.: The existence of periodic solutions for a class of differential difference equations. Chinese ann. Math. 10A, 289-294 (1989)