scholarly journals Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations

1989 ◽  
Vol 79 (402) ◽  
pp. 0-0 ◽  
Author(s):  
Hans-Otto Walther
Keyword(s):  
Periodic Solutions ◽  
Delay Equations ◽  
Differential Delay ◽  
Homoclinic Points ◽  
Differential Delay Equations

Wright's equation has no solutions of period four

1989 ◽  
Vol 113 (3-4) ◽  
pp. 281-288 ◽  
Author(s):  
Roger D. Nussbaum
Keyword(s):  
Periodic Solutions ◽  
Fourth Order ◽  
Delay Equations ◽  
Order System ◽  
Minimal Period ◽  
Differential Delay ◽  
Differential Delay Equations

SynopsisLet N:ℝ→ℝ be a locally Lipschitzian map such that (y + l)N(y)>0 for all y ≠ –1 and such that N(y)=1 + y for – 1 ≦ y ≦ 3. For any positive number α the equation y'(t) αy(t–1)N(y(t)) has, aside from the constantsolutions y(t) ≡ –1, and y(t) ≡–1 solution y(t) such that y(t + 4) = y(t) for all real t If N(y) = 1 + y for all y, one obtains Wright's equation, which isknown to have periodic solutions of minimal period p (depending on α) arbitrarily close to 4. Some results concerning nonexistence of periodic solutions of period 4 of other differential-delay equations are also proved. In all cases the method of proof consists in analysing an associated fourth-order system of ordinary differential equationsand showing that this system has no nonconstant periodic solutions.

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