Analytic Normal Forms And Limit Cycle Bifurcations Of Dynamical Systems

In this thesis, we study the algebraic results of differential systems, the global dynamics of differential systems with invariants, and the existence of analytic normal forms of analytic differential systems and diffeomorphisms. In details, we give the algebraic integrability, limit cycle bifurcation and global dynamics of generalized Lorenz systems with invariant algebraic surfaces; averaging methods of arbitrary order and its application to periodic solutions and integrability; the existence of analytic normalization of analytically integrable differential systems near a periodic orbit and the analytic normal forms of diffeomorphisms or vector fields on torus and the almost reducibility of a family of diffeomorphisms or vector fields on torus. The study of this thesis is divided into four parts:In the first part, we consider the algebraic integrability, limit cycle bifurcation and global dynamics of generalized Lorenz systems. Algebraic integrability is one of the most important research directions of dynamical systems. After Darboux [Bull. Sci.Math. 2(1878), 60–96, 123–144, 151–200]、Bruns [Theory of Differential Equations by Forsyth, 1900] and Poincar′e [Rend. Circ. Mat. Palermo 5(1981), 161–191; 11(1987),193–239], the algebraic integrability problem is reduced to give completely Darboux polynomials. Poincar′e noticed the difficulty for obtaining an algorithm to compute Darboux polynomials. The search of invariant algebraic surfaces of Lorenz systems was started from Segur [Soliton and the inverse scattering transform, in Topics in OceanPhysics, 1982], but it has not been solved until Llibre and Zhang [J. Math. Phys. 43(2002), 1622–1645], who characterized all Darboux polynomials, and consequently get the algebraic integrability of the Lorenz system.Along this direction, we will study the generalized Lorenz system˙x = a(y- x) = P(x, y, z),˙y = bx + cy- xz = Q(x, y, z),˙z = dz + xy = R(x, y, z),where x, y and z are real variables, and a, b, c and d are real parameters. Obviously,this system unifies the well-known Lorenz system [J. Atmos. Sci. 20(1963), 130–141],Chen system [Internat. J. Bifur. Chaos 9(1999), and L¨u system [Internat. J. Bifur.Chaos 12(2002), 659–661]. Using the characteristic curves for solving linear partial differential equations and the blow up techniques, we get the generators for the set of all Darboux polynomials of the generalized Lorenz system, and characterize the global dynamics of the generalized Lorenz system with an invariant algebraic surface. This result improves the results of [Llibre and Zhang, J. Math. Phys. 2002; Cao and Zhang,J. Math. Phys. 2007] in two aspects. First we provide a unified proof of this kind of systems which contains the classical Lorenz system, the Chen system and the L¨u system. Second we prove the non–existence of limit cycles of the generalized Lorenz systems and completely characterize the α and ω limits of all orbits starting from outside the invariant algebraic surface and the infinity, both of which are not solved in the previous published papers even for classical Lorenz system.In the second part, we study the theory of averaging methods of arbitrary order and its application to periodic solutions and integrability. Averaging theory is one of the best tools for studying the periodic solutions. There are many results for lower order averaging methods and it applications. Recently the averaging theory was extended to arbitrary order. Gin′e et al [Physica D 250(2013), 58–65] provided an arbitrary order averaging formula when n = 1. Llibre et al [Nonlinearity 27(2014), 2417–2417]give the arbitrary order averaging formula when the periodic orbits of the unperturbed differential system are fulled up with an n dimensional manifold. And they applied their results to the center problem for planar systems.When the periodic solutions of the unperturbed differential system ˙x = F0(t, x)form a submanifold with dimension less than n, Malkin [81] and Roseau [97] provided the averaging theory of first order. Buic?a et al [Comm. Pure Appl. Anal. 6(2007),103–111; Physica D 241(2012), 528–533] extended the Malkin and Roseau’s first order averaging theory to second order. But they cannot get the arbitrary order averaging formula in this situation, here we extend these previous results to arbitrary order for any finite dimensional periodic analytic differential system. Our results improve the previous ones in two aspects. First, without any special assumptions we give the averaging methods of arbitrary order, which contains all known results as special cases.Second, we prove a result similar to Bautin’s one for studying center–focus problem,and we can use this averaging theory to study not only the center-focus problem of planar analytic differential systems, but also if a higher dimensional differential system has periodic orbits fulled with a neighborhood of the origin. Third, we provide the formula of arbitrary order averaging methods of planar differential systems around a nilpotent singularity and some other degenerate singularities, and applied them to study the problems of integrability and limit cycle bifurcations.In the third part, we study the analytic normalization of analytically integrable differential systems near a periodic orbit. There are many results in normal form theory, which is one of the most important methods in studying dynamics of dynamical systems. Here we mainly consider the Poincar′e normal form, in which the most difficulty is to prove the convergence of the normalization transforming the original system to its Poincar′e normal form. In this direction there are some classical results such as Poincar′e–Dulac theorem, Siegel theorem and Bruno theorem. Recently, Zung et al find a connection between analytically integrable system and analytic normalization.Zung [Ann. Math. 161(2005), 141–156; Math. Res. Lett. 9(2002), 217–228] showed via torus action that any analytically integrable Hamiltonian system near a singularity is analytically equivalent to its Birkhoff normal form. Zhang [J. Differential Equations 244(2008), 1080–1092;254(2013), 3000–3022] presented a similar result using analytic methods and provided the explicit expression of the normal form.We will consider the analytic differential system˙x = f(x), x ∈ ?  Rn,where the dot denotes the derivative with respect to the time t, ? is an open subset of Rnand f(x) ∈ Cω(?). Here Cω(?) denotes the ring of analytic functions defined in ?. Assume that this system has a periodic orbit, saying Γ, located in the region ?,and it is analytically integrable in a neighborhood of Γ, then this system is analytically equivalent to its distinguished normal form in a neighborhood of the periodic orbit.This result provides a further relation between analytically integrable system and its analytic normalization. Comparing to the analytic normalization of analytically integrable system near the periodic orbit, the most difficulty of this result are: First, the number of functionally independent analytic first integrals less than the rank of linear space spanned by resonance set, we use the Floquet multiplier must has 0 eigenvalue to solve this problem; Second, in the proof of the existence of analytic normalization,we do not require the linear matrices can diagonalization, and give the general proof.In the fourth part, we study the analytic normal form of perturbed vector fields or diffeomorphisms on the torus and the almost reducibility of a family of analytic vector fields or diffeomorphisms on a torus. Relatively there are few results on normal forms of dynamical systems on tori, When the parameters of unperturbed system satisfy the Diophantine conditions, Arnold [Geometric Methods in Theory of Ordinary Differential Equations] proved that the perturbed diffeomorphisms on a torus is analytic reducible.Treschev and Zubelevich [Introduction to the Perturbation Theory of Hamiltonian System] proved a result on the reducibility of analytic perturbed vector fields on a torus to its linear part using the KAM method. P′erez–Marco [Comm. Math. Phys.223(2001), 451–464; Ann. Math. 157(2003), 557–574] proved that a family of germs of holomorphic local vector fields or diffeomorphisms near an equilibrium with the same linear part, are either all analytically reducible or not analytically reducible except for a pluripolar set.Based on their works, we prove the existence of analytic normalization for perturbed vector fields or diffeomorphisms on a torus when the parameters satisfy some conditions, and moreover we extend the P′erez–Marco’s results to almost reducibility for a family of analytic vector fields or diffeomorphisms on the torus. The difficulties to prove this extend results are: First, in order to get the analytic solution of homology equation, we need consider the Fourier theory of strip field in complex. Second, in proof of almost reducible, we construct the countable decompose of non-Pluripolar set,which is important to prove our theorem.