Analytic Integrability Of Some Systems And Related Topics

Dynamical system is an important component of the nonlinear science. It is a subject which takes research on practical problems of dynamics laws about the state changing with the time.For a long time, integrability and non-integrability has been an important problem in dynamical system's research field, and the relevant problems have been received widely attention by researchers.In the second chapter, we will introduce the formal and analytic integrability of the Lorenz system.The well-known Lorenz system can be written as Here we study the first integrals of the Lorenz system that can be described by formal power series.In particular, if s≠0 and either b is not a negative rational number, or b is a negative rational number and k1b+k2(1+s)≠0, for all k1 and k2 non-negative integers with k1+k2> 0, then the Lorenz system has no analytic first integrals in a neighborhood of the origin.Assume that s=0 and b is not a negative rational. If f=f(x, y, z) is a formal power series first integral of the Lorenz system (1), then f is a formal power series in the variable x.Now we will study the case s≠0. Since s is a parameter of the system, we can think of system (1) as the following system in the four variables x, y, z, s,Theorem 1 Suppose that s≠0 and b is not a negative rational. If f=f(x, y, z, s) is a formal power series first integral of system (2), then f is a formal power series in the variables.In the third chapter, we will introduce the analytic first integrals of the FitzHugh-Nagumo systems.We study the analytical integrability of the FitzHugh-Nagumo systems with parameters a, b,c,d∈(?).Theorem 2 For the FitzHugh-Nagumo system (3) with b=0 the following statements hold.(a) If c=0 then system (3) is integrable with the global analytic first integralsΦ1=y and(b) If c≠0 then the unique formal series first integral of system (3) in a neighborhood of the singular point is of the form f(y) where f is an arbitrary formal series in the variabley.(c) If c≠0 then the unique global analytic first integrals of system (3) are of the form f(y) where f is an arbitrary global analytic function.Theorem 3 The following statements hold for the FitzHugh-Nagumo system (3) sys-tem withb≠0.(a) If c≠0,it has no global analytic first integrals which are analytic in the parameter b in a neighborhood of b=0.(b) If c=0,it has no global analytic first integrals which are analytic in the parameter b in a neighborhood of b=0.Theorem 4 Assume that the FitzHugh-Nagumo system (3) with b≠0, c=0 and d≠-1/a where a≠0 satisfies one of the following three conditions(a) 4a3+b2=0,Then the system has no analytic first integrals in a neighborhood of the singular point (0,0,0).In the forth chapter, we will introduce the global analytic integrability of the Rabinovich system.Theorem 5 The following statements hold for the Rabinovich system withb3=0.(a) If b1b2≠0, b1+b2≠0, it has no global analytic first integrals. (b) If h≠0,b2=-b1=-b≠0, it has no global analytic first integrals which are also analytic in the parameters b and h in a neighborhood of (b, h)= (0,0).(c) If h=0, b2=-b1=-b≠0, it has no global analytic first integrals which are also analytic in the parameters b in a neighborhood of b=0.(d) Let H1=x2+y2-4hz, H2=y2+z2-2hz. For any other values of the parameters (h, b1,b2) its unique global analytic first integrals are of the form;(d.1) g(H1, H2) if b1+b2=0, where g(·,·) is an arbitrary global analytic function.(d.2) g(H1-H2) if b1=0, b2≠0, where g(·) is an arbitrary global analytic function.(d.3) g(H2) if b1≠0, b2=0, where g(·) is an arbitrary global analytic function.Theorem 6 The following statements hold for the Rabinovich system with b3≠0.(a)Ifb1+b2≠0, it has no global analytic first integrals which are also analytic in the parameters b3 in a neighborhood of b3=0.(b) If h≠0, b2=-b1=-b≠0, it has no global analytic first integrals which are also analytic in the parameters h, b and b3 in a neighborhood of (h, b, b3)=(0,0,0).(c) If h=0, b2=-b1=-b≠0, it has no global analytic first integrals which are also analytic in the parameters b and b3 in a neighborhood of (b, b3)=(0,0).(d)For any other values of the parameters h, b1,b2, its unique global analytic first inte-gral are of the form g(H1), where g(·) is an arbitrary global analytic function with H1=x2+y2 if h＝b1=b2＝0.