| Many mathematicians have studied the existence of periodic solutions for various delay diferential equations,which is important both in theory and in practice.Fixed point theorems are the principal tools to investigate the existence of such solutions. In 1974, Kaplan and Yorke studied and introduced a new technique for estab- lishing the existence of periodic solutions for certain special classes of delay diferential equations by studying a related system of coupled ordinary difer- ential equations. From then on, there have appeared many works concerning the existence of periodic so-luations of scalar diferential equations.In our paper,we will do this here for four different kind of systems of delay differential equations.(x|·)(t)=f(y(t-s))g(z(t-s))+f(z(t-s))g(y(t-s)), (y|·)(t)=-f(x(t-s))g(z(t-s))+f(z(t-s))g(x(t-s)), (z|·)(t)=-f(x(t-s))g(y(t-s))-f(y(t-s))g(x(t-s)); (1)(x|·)i(t)=-∑j=1k aijf(xj(t-s))-∑j=k+1nf(xj(t-s)),i≤k,(x|·)i(t)=∑j=1k f(xj(t-s))-∑j=k+1n aijf(xj(t-s)),k11) f∈Cr,r≥3,f(-x) = -f(x),xf(x) > 0 for 0 < x < A.where A is a constant,and f'(0)=ω>0;(H21) g∈Cr,r≥2,g(-x) = g(x),g(0) =α> 0,and g(x) > 0 for 0 < x < A. Then the following system(x|·)(t)=f(y)g(z)+f(z)g(y),(y|·)(t)=-f(x)g(z)+f(z)g(x), (z|·)(t)=-f(x)g(y)-f(y)g(x).has a family of periodic solutions, which lies in the symplectic leaf C(x, y, z)=0,surrounding the origin and also contained in this leaf,and forms an origin'speriod annulus with the period function P(h). Moreover, if(H31) P(h) takes the value 2sthen system (1)has a periodic orbit of period 2s.Theorem 2. Suppose that system(2)satisfies(H12)f∈C1(R) for x≠0, f(-x) = -f(x), f(0) = 0, xf(x) > 0,and f'(x) > 0 for all x≠0;(H22)limx→0[f(x)/x]=f'(0)=ω0,limx→∞[f(x)/x]=ω∞hereω0 andω∞is nozero constants; (H32)There exists a constant p>0 satisfyingω∞<(2πj)/p cot(((n-1)π)/(2n))<(2πj)/p cot(π/(2n))<ω0, n=2k,ω∞<(2πj)/p cot(((n-1)π)/(2n))<(2πj)/p cot(π/n)<ω0, n=2k+1.where j≤1 is an integer; (H42)p(?)(2π/ω∞cot((2q+1)π/(2n)))N (q=0,1,2,…,n/2-1),n = 2k,p(?)(2π/ω∞cot(qπ/n)N (q=0,1,2,…,(n-1)/2),n=2k+1.where N denote the set of positive integers. Then system (2) has a periodic orbit of period p = 2s.Theorem 3. Suppose that system(3) satisfiesF∈C1,F(x, y, -x) = -F(-x, -y, x).If the following differential system(x|·) = F(x, y, -x), y = F(y, -x, -y).has a periodic orbitγwith period (?) = 2s which surrounds the origin,then the delay differential system (3) has a periodic orbit with period (?) = 2s.Theorem4. Suppose that system (4) satisfiesF∈Cl withl≥3,F(-x, -y, -z) = -F(x, y, z),and F(x, x + z,z) - F(x + z, z, -x) - F(-z, x,x + z) = 0.If the following differential system(x|·) = -F(x, y, z), (y|·) = -F(y, z, -x), (z|·) = -F{z, -x, -y).has a periodic orbitγwith period(?) = 2s which starts at(0,a,a)∈G = x - y + z = 0 and surrounds the origin on G,then the delay differential system (4)has a periodic orbit with period (?)=2s. For more we stady the system(?)=f(y(t-r2))g(z(t-r3))+f(z(t-r3))g(y(t-r2)), (?)=-f(x(t-r1))g(z(t-r3))+f(z(t-r3))g(x(t-r1)), (?)=-f(x(t-r1))g(y(t-r2))-f(y(t-r2))g(x(t-r1)); (5)Theorem5. Suppose that system (5) satisfies (H11), (H21), (H31),if also (H41)hold.And there are k1, k2, k3 so thatr1/(2k1+1)=r2/(2k2+1)=r3/(2k3+1)=μthen(5) has a periodic orbit with period 2μ.Then we stady the system(x|·)(t)=-af(v(t-s))g(z(t-s))-bf(y(t-s))g(u(t-s))+af(z(t-s))g(y(t-s))-cf(z(t-s))g(u(t-s))+bf(u(t-s))g(y(t-s))+cf(u(t-s))g(z(t-s)),(y|·)(t)=af(x(t-s))g(u(t-s))+bf(x(t-s))g(u(t-s))-af(z(t-s))g(x(t-s))-df(z(t-s))g(u(t-s))-bf(u(t-s))g(x(t-s))+df(u(t-s))g(z(t-s)),(z|·)(t)=-af(x(t-s))g(y(t-s))+cf(x(t-s))g(u(t-s))+af(y(t-s))g(u(t-s))+df(y(t-s))g(u(t-s))-cf(u(t-s))g(x(t-s))-df(u(t-s))g(y(t-s)),(u|·)(t)=-bf(x(t-s))g(y(t-s))-cf(x(t-s))g(z(t-s))+bf(y(t-s))g(x(t-s))-df(z(t-s))g(x(t-s))+cf(z(t-s))g(x(t-s))+df(z(t-s))g(y(t-s));(6) and its related system is(x|·)(t)=af(y)g(z)+bf(y)g(u)-af(z)g(y)+cf(z)g(u)-bf(u)g(y)-cf(u)g(z),(y|·)(t)=-af(x)g(z)-bf(x)g(u)+af(z)g(x)+df(z)g(u)+bf(u)g(x)-df(u)g(z), (z|·)(t)=af(x)g(y)-cf(x)g(u)-af(y)g(u)-df(y)g(u)+cf(u)g(x)+df(u)g(x), (u|·)(t)=bf(x)g(y)+cf(x)g(z)-bf(y)g(x)+df(z)g(x)-cf(z)g(x)-df(z)g(y). (7)Theorem6. Suppose that system(6)satisfies the following conditions(H13) f∈Cr,r≥3,f(-x)=-f(x),xf(x) > 0 for 0 < x < A.where A is a constant,and f'(0)=ω> 0;(H23) g∈Cr,r≥2,g(-x)=g(x),g(0)=α>0 and g(x)≠0 where 0 < x < A;(H33)for all 043)The setΩ={H(x, y, z, u)=0}∩{C(x, y, z, u)=0} is a 2-dimensional manifold and for the flow of system ,(7) restricted toΩthere exists a first integral K;(H53) (a, b, c, d)≠(-d, d,-d, d) for alld∈Rthen system (7)has a family of periodic orbits, which lies inΩ,near the origin and forms a period annulus with period function P(k),where P(k) denotes the period of the periodic orbit K=k.In addition, if(H63) P(k) takes the value 2s then system (6)has a periodic orbit with period 2s.
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