This paper is concerned with the boundedness of solutions for second order differential equations x + f(x, t)x + g(x, t) = 0, which are neither dissipative nor conservative, and where the functions f and g are odd in ...This paper is concerned with the boundedness of solutions for second order differential equations x + f(x, t)x + g(x, t) = 0, which are neither dissipative nor conservative, and where the functions f and g are odd in x and even in t, which are 1-periodic in t, and the function g satisfies g(x,t/x+, as|x| - +. Using the KAM theory for reversible systems, the author proves the existence of invariant tori and thus the boundedness of all the solutions and the existence of quasiperiodic solutions and subharmonic solutions.展开更多
It is shown that all solutions are bounded for Duffing equation s＋ x^2n＋1＋j=0∑^2n Pj（t）x^j=0,provided that for each n ＋ 1≤j≤2n,Pj ∈ C^γ（T^1） with γ〉1-1/n and for each j with 0 ≤ j ≤ n,Pj ∈L （T^1） w...It is shown that all solutions are bounded for Duffing equation s＋ x^2n＋1＋j=0∑^2n Pj（t）x^j=0,provided that for each n ＋ 1≤j≤2n,Pj ∈ C^γ（T^1） with γ〉1-1/n and for each j with 0 ≤ j ≤ n,Pj ∈L （T^1） where T^1=R/Z.展开更多
In this paper, we prove that the second order differential equation d^2x/dt^2＋x^2n_1f（x）＋p（t）=0with p（t ＋ 1） = p（t）, f（x ＋ T） = f（x） smooth and f（x） 〉 0, possesses Lagrangian stability despite of th...In this paper, we prove that the second order differential equation d^2x/dt^2＋x^2n_1f（x）＋p（t）=0with p（t ＋ 1） = p（t）, f（x ＋ T） = f（x） smooth and f（x） 〉 0, possesses Lagrangian stability despite of the fact that the monotone twist condition is violated.展开更多
We are concerned with the boundedness of all the solutions for second order differential equation $$\ddot x + f\left( x \right)\dot x + g\left( x \right) = e\left( t \right),$$ , wheref(x) andg(x) are odd, e( t) is od...We are concerned with the boundedness of all the solutions for second order differential equation $$\ddot x + f\left( x \right)\dot x + g\left( x \right) = e\left( t \right),$$ , wheref(x) andg(x) are odd, e( t) is odd and 1-periodic, andg(x) satisfies $$Sign \left( x \right) \cdot g\left( x \right) \to + \infty ,\frac{{g\left( x \right)}}{x} \to 0,as\left| x \right| \to + \infty .$$展开更多
In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s p...In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.展开更多
In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity x'' + V'(x) + g(x) = e(t, x, x'),where the assumptions on V, g a...In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity x'' + V'(x) + g(x) = e(t, x, x'),where the assumptions on V, g and e are regular, described precisely in the introduction.Using a variant of Moser's twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.展开更多
In this paper, we are concerned with the boundedness of all the solutions ofthe equation x' + ax^+ - bx^- + φ(x) = p(t), where p(t) is a smooth 2π-periodic function, a and bare positive constants, and the pertur...In this paper, we are concerned with the boundedness of all the solutions ofthe equation x' + ax^+ - bx^- + φ(x) = p(t), where p(t) is a smooth 2π-periodic function, a and bare positive constants, and the perturbation φ(x) is bounded.展开更多
文摘This paper is concerned with the boundedness of solutions for second order differential equations x + f(x, t)x + g(x, t) = 0, which are neither dissipative nor conservative, and where the functions f and g are odd in x and even in t, which are 1-periodic in t, and the function g satisfies g(x,t/x+, as|x| - +. Using the KAM theory for reversible systems, the author proves the existence of invariant tori and thus the boundedness of all the solutions and the existence of quasiperiodic solutions and subharmonic solutions.
基金Project supported by the National Natural Science Foundation of China(No.11421061)
文摘It is shown that all solutions are bounded for Duffing equation s＋ x^2n＋1＋j=0∑^2n Pj（t）x^j=0,provided that for each n ＋ 1≤j≤2n,Pj ∈ C^γ（T^1） with γ〉1-1/n and for each j with 0 ≤ j ≤ n,Pj ∈L （T^1） where T^1=R/Z.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10871142, 10871090)
文摘In this paper, we prove that the second order differential equation d^2x/dt^2＋x^2n_1f（x）＋p（t）=0with p（t ＋ 1） = p（t）, f（x ＋ T） = f（x） smooth and f（x） 〉 0, possesses Lagrangian stability despite of the fact that the monotone twist condition is violated.
基金The author is very grateful to Professors Ding Tongren and Liu Bin for their valuable suggestions for this paper.
文摘We are concerned with the boundedness of all the solutions for second order differential equation $$\ddot x + f\left( x \right)\dot x + g\left( x \right) = e\left( t \right),$$ , wheref(x) andg(x) are odd, e( t) is odd and 1-periodic, andg(x) satisfies $$Sign \left( x \right) \cdot g\left( x \right) \to + \infty ,\frac{{g\left( x \right)}}{x} \to 0,as\left| x \right| \to + \infty .$$
文摘In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.
基金supported by the National Natural Science Foundation of China(No.10325103)the Chinese Scholarship Council(No.201206010092)
文摘In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity x'' + V'(x) + g(x) = e(t, x, x'),where the assumptions on V, g and e are regular, described precisely in the introduction.Using a variant of Moser's twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.
文摘In this paper, we are concerned with the boundedness of all the solutions ofthe equation x' + ax^+ - bx^- + φ(x) = p(t), where p(t) is a smooth 2π-periodic function, a and bare positive constants, and the perturbation φ(x) is bounded.
