Tian Jinghuang

Bio: Tian Jinghuang is an academic researcher from Chinese Academy of Sciences. The author has contributed to research in topics: Quadratic equation. The author has an hindex of 1, co-authored 1 publications receiving 60 citations.

On General Properties of Quadratic Systems

Abstract: (1982). On General Properties of Quadratic Systems. The American Mathematical Monthly: Vol. 89, No. 3, pp. 167-178.

Ordinary Differential Equations with Applications

Abstract: Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

Hilbert's 16th problem and bifurcations of planar polynomial vector fields

Abstract: The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert's 16th problem? Section 2: The first part of Hilbert's 16th problem. Section 3: The second part of Hilbert's 16th problem: introduction. Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert's 16th problem. Section 7: Global and local bifurcations of Zq–equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.

Non-autonomous equations related to polynomial two-dimensional systems

Abstract: Periodic solutions of certain one-dimensional non-autonomous differential equations are investigated (equation (1.4)); the independent variable is complex. The motivation, which is explained in the introductory section, is the connection with certain polynomial two-dimensional systems. Several classes of coefficients are considered; in each case the aim is to estimate the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. In particular, we need to know when there is a full neighbourhood of periodic solutions. We give a number of sufficient conditions and investigate the implications for the corresponding two-dimensional systems.

The number of limit cycles of certain polynomial differential equations

Abstract: Two-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

A new dynamical approach of Emden-Fowler equations and systems

Abstract: We give a new approach on general systems of the form \[ (G)\left\{ \begin{array} [c]{c}% -\Delta_{p}u=\operatorname{div}(\left\vert abla u\right\vert ^{p-2} abla u)=\varepsilon_{1}\left\vert x\right\vert ^{a}u^{s}v^{\delta},\\ -\Delta_{q}v=\operatorname{div}(\left\vert abla v\right\vert ^{q-2} abla u)=\varepsilon_{2}\left\vert x\right\vert ^{b}u^{\mu}v^{m}, \end{array} \right. \] where $Q,p,q,\delta,\mu,s,m,$ $a,b$ are real parameters, $Q,p,q eq1,$ and $\varepsilon_{1}=\pm1,$ $\varepsilon_{2}=\pm1.$ In the radial case we reduce the problem to a quadratic system of order 4, of Kolmogorov type. Then we obtain new local and global existence or nonexistence results. In the case $\varepsilon_{1}=\varepsilon_{2}=1,$ we also describe the behaviour of the ground states in two cases where the system is variational. We give an important result on existence of ground states for a nonvariational system with $p=q=2$ and $s=m>0.$ In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$ and $\delta=m+1$ and $\mu=s+1.$