On the use of semi-invariants for the stability analysis of planar systems

This paper is concerned with the concept of semi-invariant and its use for the stability analysis of the origin for nonlinear systems of order two. Well known tools from differential geometry such as (orbital) symmetries render the proposed results quite interesting from a computational point of view. The proper connections with center manifold theory are pointed out.

[1]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[2]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[3]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[4]  H. Stephani.  Differential Equations: Their Solution Using Symmetries , 1990 .

[5]  J. P. LaSalle,et al.  Recent Advances in Liapunov Stability Theory , 1964 .

[6]  Henry Hermes,et al.  Nilpotent and High-Order Approximations of Vector Field Systems , 1991, SIAM Rev..

[7]  G. Gaeta.  Poincaré Normal and Renormalized Forms , 2001, math-ph/0106027.

[8]  Nahum Shimkin,et al.  Nonlinear Control Systems , 2008 .

[9]  A. Goriely.  Integrability and Nonintegrability of Dynamical Systems , 2001 .

[10]  G. Belitskii.  C∞-Normal Forms of Local Vector Fields , 2002 .

[11]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[12]  A. Fuller,et al.  Stability of Motion , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[13]  J. Meinhardt,et al.  Symmetries and differential equations , 1981 .

[14]  G. Cicogna,et al.  Symmetry and Perturbation Theory in Nonlinear Dynamics , 1999 .

[15]  S. Walcher.  Plane polynomial vector fields with prescribed invariant curves , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  C. Christopher.  Invariant algebraic curves and conditions for a centre , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[17]  L. Rosier.  Homogeneous Lyapunov function for homogeneous continuous vector field , 1992 .

[18]  A. Bacciotti,et al.  Liapunov functions and stability in control theory , 2001 .

[19]  M. Kawski.  Homogeneous Stabilizing Feedback Laws , 1990 .

[20]  Zongli Lin,et al.  Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Information Sciences 267 , 2002 .

[21]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[22]  M. Kawski.  GEOMETRIC HOMOGENEITY AND STABILIZATION , 1995 .

[23]  C. Moog,et al.  Algebraic Methods for Nonlinear Control Systems , 2006 .

[24]  Andrea Bacciotti,et al.  Local Stabilizability of Nonlinear Control Systems , 1991, Series on Advances in Mathematics for Applied Sciences.

[25]  Giuseppe Gaeta,et al.  Symmetry invariance and centre manifolds for dynamical systems , 1994 .