Abstract
We are describing the stable nonautonomous planar dynamic systems with complex coefficients by using the asymptotic solutions (phase functions) of the characteristic (Riccati) equation. In the case of nonautonomous dynamic systems, this approach is more accurate than the eigenvalue method. We are giving a new construction of the energy (Lyapunov) function via phase functions. Using this energy, we are proving new stability and instability theorems in terms of the characteristic function that depends on unknown phase functions. By different choices of the phase functions, we deduce stability theorems in terms of the auxiliary function of coefficients RA(t), which is invariant with respect to the lower triangular transformations. We discuss some examples and compare our theorems with the previous results. MSC: 34D20
Highlights
1 Introduction We are interested in the behavior of a given solution u(t) of the nonlinear planar dynamic system u (t) = A(t, u)u(t), A(t, u) = a (t, u(t)) a (t, u(t)), t ≥ T, ( . )
This example shows that the description of stability of nonautonomous dynamic systems in terms of the eigenvalues is not accurate
We prove main stability theorems for two-dimensional systems in terms of unknown phase functions
Summary
This example shows that the description of stability of nonautonomous dynamic systems in terms of the eigenvalues is not accurate. Using this energy, we prove main stability theorems for two-dimensional systems in terms of unknown phase functions To show that our theorems are useful, we deduce different versions of stability theorems (old well-known and some new ones) by using different phase functions as asymptotic solutions of the characteristic equation ) since there is no universal formula for an asymptotic solution of the characteristic equation. Using these theorems we derive the versions of stability theorem of Pucci-Serrin [ ], Smith [ ], and some new ones. For diagonal system ( . ), formulas ( . ) fail (for this case, see (A. ))
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
