Bifurcation from a heteroclinic solution in differential delay equations
Author:
HansOtto Walther
Journal:
Trans. Amer. Math. Soc. 290 (1985), 213233
MSC:
Primary 34K15; Secondary 58F22
MathSciNet review:
787962
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Abstract: We study a class of functional differential equations with periodic nonlinearity in and in . Such equations describe a state variable on a circle with one attractive rest point (given by the argument of ) and with reaction lag to deviations. We prove that for a certain critical value there exists a heteroclinic solution going from the equilibrium solution to the equilibrium . For , this heteroclinic connection is destroyed, and periodic solutions of the second kind bifurcate. These correspond to periodic rotations on the circle.
 [1]
A.
A. Andronow and C.
E. Chaikin, Theory of Oscillations, Princeton University
Press, Princeton, N. J., 1949. English Language Edition Edited Under the
Direction of Solomon Lefschetz. MR 0029027
(10,535f)
 [2]
Nathaniel
Chafee, A bifurcation problem for a functional differential
equation of finitely retarded type, J. Math. Anal. Appl.
35 (1971), 312–348. MR 0277854
(43 #3587)
 [3]
Shui
Nee Chow and Jack
K. Hale, Methods of bifurcation theory, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Science], vol. 251, SpringerVerlag, New YorkBerlin, 1982. MR 660633
(84e:58019)
 [4]
ShuiNee
Chow and John
MalletParet, Singularly perturbed delaydifferential
equations, Coupled nonlinear oscillators (Los Alamos, N. M., 1981)
NorthHolland Math. Stud., vol. 80, NorthHolland, Amsterdam, 1983,
pp. 7–12. MR
742005, http://dx.doi.org/10.1016/S03040208(08)70968X
 [5]
Tetsuo
Furumochi, Existence of periodic solutions of onedimensional
differentialdelay equations, Tôhoku Math. J. (2)
30 (1978), no. 1, 13–35. MR 0470418
(57 #10172)
 [6]
Jack
K. Hale, Functional differential equations, Analytic theory of
differential equations (Proc. Conf., Western Michigan Univ., Kalamazoo,
Mich., 1970) Springer, Berlin, 1971, pp. 9–22. Lecture Notes
in Mat., Vol. 183. MR 0390425
(52 #11251)
 [7]
Jack
Hale, Theory of functional differential equations, 2nd ed.,
SpringerVerlag, New YorkHeidelberg, 1977. Applied Mathematical Sciences,
Vol. 3. MR
0508721 (58 #22904)
 [8]
Jack
K. Hale and Krzysztof
P. Rybakowski, On a gradientlike integrodifferential
equation, Proc. Roy. Soc. Edinburgh Sect. A 92
(1982), no. 12, 77–85. MR 667126
(84j:45040), http://dx.doi.org/10.1017/S0308210500019946
 [9]
J. MalletParet and R. D. Nussbaum, Global continuation and asymptotic behavior of periodic solutions of differentialdelay equations, in preparation.
 [10]
L.
P. Šil′nikov, The generation of a periodic motion from
a trajectory which is doubly asymptotic to a saddle type equilibrium
state, Mat. Sb. (N.S.) 77 (119) (1968), 461–472
(Russian). MR
0255922 (41 #582)
 [1]
 A. A. Andronov and C. E. Chaikin, Theory of oscillations, Princeton Univ. Press, Princeton, N.J., 1949. MR 0029027 (10:535f)
 [2]
 N. Chafee, A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl. 35 (1971), 312348. MR 0277854 (43:3587)
 [3]
 S. N. Chow and J. K. Hale, Methods of bifurcation theory, Springer, New York, Heidelberg and Berlin, 1982. MR 660633 (84e:58019)
 [4]
 S. N. Chow and J. MalletParet, Singularly perturbed delaydifferential equations, Coupled Nonlinear Oscillators (J. Chandra and A. C. Scott, eds.), NorthHolland Math. Studies, vol. 80, 1983, pp. 712. MR 742005
 [5]
 T. Furumochi, Existence of periodic solutions of onedimensional differentialdelay equations, Tôhoku Math. J. 30 (1978), 1335. MR 0470418 (57:10172)
 [6]
 J. K. Hale, Functional differential equations, Springer, New York, Heidelberg and Berlin, 1971. MR 0390425 (52:11251)
 [7]
 , Theory of functional differential equations, Springer, New York, Heidelberg and Berlin, 1977. MR 0508721 (58:22904)
 [8]
 J. K. Hale and K. Rybakowski, On a gradientlike integrodifferential equation, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), 7785. MR 667126 (84j:45040)
 [9]
 J. MalletParet and R. D. Nussbaum, Global continuation and asymptotic behavior of periodic solutions of differentialdelay equations, in preparation.
 [10]
 L. P. Sil'nikov, On the generation of periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Math. USSRSb. 6 (1968); English transl., Mat. Sb. 77(119) (1968), 427438. MR 0255922 (41:582)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507879624
PII:
S 00029947(1985)07879624
Keywords:
Characteristic equation,
saddle point property,
heteroclinic solution,
Poincaré map,
periodic solution of the second kind
Article copyright:
© Copyright 1985
American Mathematical Society
