The main purpose of developing stability theory is to examine dynamic responses of a system to disturbances as the time approaches infinity. It has been and still is the object of intense investigations due to its intrinsic interest and its relevance to all practical systems in engineering, finance, natural science and social science. This monograph provides some state-of-the-art expositions of major advances in fundamental stability theories and methods for dynamic systems of ODE and DDE types and in limit cycle, normal form and Hopf bifurcation control of nonlinear dynamic systems. Presents comprehensive theory and methodology of stability analysis Can be used as textbook for graduate students in applied mathematics, mechanics, control theory, theoretical physics, mathematical biology, information theory, scientific computation Serves as a comprehensive handbook of stability theory for practicing aerospace, control, mechanical, structural, naval and civil engineers

Stability of dynamical systems

In this paper, the existence of 12 small limit cycles is proved for cubic order Z2-equivariant vector fields, which bifurcate from fine focus points. This is a new result in the study of the second part of the 16th Hilbert problem. The system under consideration has a saddle point, or a node, or a focus point (including center) at the origin, and two weak focus points which are symmetric about the origin. It has been shown that the system can exhibit 10 and 12 small limit cycles for some special cases. Further studies are given in this paper to consider all possible cases, and prove that such a Z2-equivariant vector field can have maximal 12 small limit cycles. Fourteen or sixteen small limit cycles, as expected before, are not possible.

Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields

In this paper, we introduce some new results on the study of Z
 q
 -equivariant planar polynomial vector fields. The main conclusions are as follows. (1) For the Z
 2-equivariant planar cubic systems having two elementary foci, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z
 2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme $${\langle 6\amalg 6\rangle}$$
 is proved. (2) On the basis of mentioned work in (1), by considering the bifurcation of a global limit cycle from infinity, we show that under small Z
 2-equivariant cubic perturbations, such bi-center cubic system has at least 13 limit cycles with the scheme $${\langle 1 \langle 6\amalg 6\rangle\rangle}$$
 , i.e., we obtain that the Hilbert number H(3) ≥ 13. (3) For the Z
 5-equivariant planar polynomial vector field of degree 5, we shown that such system has at least five symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z
 5-equivariant perturbations, the conclusion that perturbed system has at least 25 limit cycles with the scheme $${\langle 5\amalg 5\amalg 5\amalg 5\amalg 5\rangle}$$
 is rigorously proved. (4) For the Z
 6-equivariant planar polynomial vector field of degree 5, we proved that such system has at least six symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z
 6-equivariant perturbations, the conclusion that perturbed system has at least 24 limit cycles with the scheme $${\langle 4\amalg 4\amalg 4\amalg 4\amalg 4\amalg4\rangle}$$
 is rigorously proved. Two schemes of distributions of limit cycles are given.

New Results on the Study of Zq-Equivariant Planar Polynomial Vector Fields

In this paper, we study bifurcation of limit cycles from the equator of piecewise polynomial systems with no singular points at infinity. We develop a method for computing the Lyapunov constants at infinity of piecewise polynomial systems. In particular, we consider cubic piecewise polynomial systems and study limit cycle bifurcations in the neighborhood of the origin and infinity. Moreover, an example is presented to show 11 limit cycles bifurcating from infinity.

Bifurcation of limit cycles at infinity in piecewise polynomial systems

A System of Two Differential Equations

The computation of period constants is a way to study isochronous center for polynomial differential systems. In this article, a new method to compute period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As an application, we discuss the center conditions and isochronous centers for a class of high-degree system.

A new method to determine isochronous center conditions for polynomial differential systems

In this paper, the bifurcation of limit cycles for a cubic polynomial system is investigated By the computation of the singular point values, we prove that the system has 12 small amplitude limit cycles The process of the proof is algebraic and symbolic

A cubic system with twelve small amplitude limit cycles

The center problem and bifurcations of limit cycles for the Lyapunov system are continuously studied. We shall prove that we can construct successively a formal series such that the Lyapunov system is reduced a half-normal form. From the coefficients of the half-normal form, we obtain directly the Lyapunov constants of the origin. As examples, for two classes of cubic systems, the center and focus problem, and multiple bifurcations of limit cycles are studied.

New study on the center problem and bifurcations of limit cycles for the lyapunov system (ii)

In this paper, Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems is studied, a new algorithm of the formal series for the flow on center manifold is discussed, from this, a recursion formula for computation of the singular point quantities is obtained for the corresponding bifurcation equation, which is linear and then avoids complex integrating operations, therefore the calculation can be readily done with using computer symbol operation system such as Mathematica, and more the algebraic equivalence of the singular point quantities and corresponding focal values is proved, thus Hopf bifurcation can be considered easily. Finally an example is studied, by computing the singular point quantities and constructing a bifurcation function, the existence of 5 limit cycles bifurcated from the origin for the flow on center manifold is proved.

Yirong Liu

Papers

A new method to determine isochronous center conditions for polynomial differential systems

A cubic system with twelve small amplitude limit cycles

New study on the center problem and bifurcations of limit cycles for the lyapunov system (ii)

Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems

Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields