This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+...This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+a_(4)xy^(3)+a_(5)xy^(4)=xσ(y),and any zero of 1+a_(1)y+a_(2)y^(2)+a_(3)y^(3)+a_(4)y^(4)+a_(5)y^(5),y=y is an invariant straight line.At last,all global phase portraits are drawn on the Poincare disk.展开更多
In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter cond...In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.展开更多
First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle： {x ∈ R^n ： 0 ≤ xi ≤ ki, i = 1,..., n} still holds. Next, based on the theore...First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle： {x ∈ R^n ： 0 ≤ xi ≤ ki, i = 1,..., n} still holds. Next, based on the theorem a competitive system with the linear structure saturation defined on the n-rectangle is investigated, which admits a unique （n - 1）- dimensional carrying simplex as a global attractor. Further, we focus on the whole dynamical behavior of the three-dimensional case, which has a unique locally asymptotically stable positive equilibrium. Hopf bifurcations do not occur. We prove that any limit set is either this positive equilibrium or a limit cycle. If limit cycles exist, the number of them is finite. We also give a criterion for the positive equilibrium to be globally asymptotically stable.展开更多
基金supported by National Natural Science Foundation of China(No.12301197)Natural Science Foundation of Henan(No.232300420343)+2 种基金Science and Technology Research Project of Henan Province(No.232102210057)Scientific Research Foundation for Doctoral Scholars of Haust(No.13480077)Natural Science Foundation of Hunan(No.2021JJ30166)。
文摘This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+a_(4)xy^(3)+a_(5)xy^(4)=xσ(y),and any zero of 1+a_(1)y+a_(2)y^(2)+a_(3)y^(3)+a_(4)y^(4)+a_(5)y^(5),y=y is an invariant straight line.At last,all global phase portraits are drawn on the Poincare disk.
基金supported by Shanghai Leading Academic Discipline Project (No. S30501)Shanghai Natural Science Foundation Project (No. 10ZR1420800)
文摘In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.
文摘First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle： {x ∈ R^n ： 0 ≤ xi ≤ ki, i = 1,..., n} still holds. Next, based on the theorem a competitive system with the linear structure saturation defined on the n-rectangle is investigated, which admits a unique （n - 1）- dimensional carrying simplex as a global attractor. Further, we focus on the whole dynamical behavior of the three-dimensional case, which has a unique locally asymptotically stable positive equilibrium. Hopf bifurcations do not occur. We prove that any limit set is either this positive equilibrium or a limit cycle. If limit cycles exist, the number of them is finite. We also give a criterion for the positive equilibrium to be globally asymptotically stable.
