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A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the difference equation. The renormalization group equation is a Lie differential equation of a Lie group which leaves the system approximately invariant. For a 2-D symplectic map, the renormalization group equation becomes a Hamiltonian system and a long-time behaviour of the symplectic map is described by the Hamiltonian. We study the Poincar\'e-Birkoff bifurcation in the 2-D symplectic map by means of the Hamiltonian and give a condition for the bifurcation., Comment: Accepted to J. Phys. A, 7 pages

A renormalization group method with the Lie symmetry is presented for the singular perturbation problems. Asymptotic solutions are obtained as group-invariant solutions under approximate Lie group admitted by perturbed differential equations., Comment: 9 pages, 1 figure

A method of calculating a new class of symmetries is presented for partial differential equations. The method give a new dynamical solution for an isothermal and cylindrically symmetric hydrodynamics equations under self-gravity. The solution describes a development of a gas cylinder from a motion-less state to states of expansion and oscillation.

A new dynamical solution for a gas sphere under self-gravity is presented to describe a development of a gas sphere from a motion-less state to a state of expansion with a constant speed and a reflection phenomenon in the dynamics of the surface of the sphere., Comment: 6 pages, 4figures

A symplecticity-preserving RG analysis is carried out to study a resonance structure near an elliptic fixed point of a proto-type symplectic map in two dimensions. Through analyzing fixed points of a reduced RG map, a topology of the resonance structure such as a chain of resonant islands can be determined analytically. An application of this analysis to the Henon map is also presented., Comment: 10 pages

We present a method to construct symplecticity-preserving renormalization group maps by using the Liouville operator, and obtain correctly reduced symplectic maps describing their long-time behavior even when a resonant island chain appears.

We present various exact solutions of a discrete complex Ginzburg-Landau (CGL) equation on a plane lattice, which describe target patterns and spiral patterns and derive their stability criteria. We also obtain similar solutions to a system of van der Pol's oscillators., Comment: Latex 11 pages, 17 eps files

We present a method to construct a symplecticity preserving renormalization group map of a chain of weakly nonlinear symplectic maps and obtain a general reduced symplectic map describing its long-time behaviour. It is found that the modulational instability in the reduced map triggers random wandering of orbits around some homoclinic-like manifolds, which is understood as the Bernoulli shifts., Comment: submitted to Prog. Theor. Phys

By means of the perturbative renormalization group method, we study a long-time behaviour of some symplectic discrete maps near elliptic and hyperbolic fixed points. It is shown that a naive renormalization group (RG) map breaks the symplectic symmetry and fails to describe a long-time behaviour. In order to preserve the symplectic symmetry, we present a regularization procedure, which gives a regularized symplectic RG map describing an approximate long-time behaviour succesfully.

By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative H\'enon map, both of which exhibit the strong homoclinic chaos. In terms of the asymptotic expansion, a simple formulation is presented to give the first homoclinic point in the double-well map. Even a truncated expansion of the unstable manifold is shown to reproduce the well-known many-leaved (fractal) structure of the strange attractor in the H\'enon map., Comment: 14pages, 7figures

An approximate Poincare map near equally strong multiple resonances is reduced by means the method of averaging. Near the resonance junction of three degrees of freedom, we find that some homoclinic orbits ``whiskers'' in single resonance lines survive and form nearly periodic orbits, each of which looks like a pair of homoclinic orbits., Comment: 14pages,2figures

We asymptotically compute the intersection angles between N-dimensional stable and unstable manifolds in 2N-dimensional symplectic mappings. There exist particular 1-dimensional stable and unstable sub-manifolds which experience exponentially small splitting of separatrix in our models. We show that the angle between the sub-manifolds is exponentially small with respect to the perturbation parameter $\epsilon$, and the other angles are $O(\epsilon^2)$., Comment: 8 pages, LaTeX

The Lie-group approach to the perturbative renormalization group (RG) method is developed to obtain an asymptotic solutions of both autonomous and non-autonomous ordinary differential equations. Reduction of some partial differetial equations to typical RG equations is also achieved by this approach and a simple recipe for providing RG equations is presented., Comment: 32pages, 2figures

Homoclinic bifurcation of 4-dimensional symplectic mappings is asymptotically studied. We construct the 2-dimensional stable and unstable manifolds near the sub-manifolds which experience exponentially small splitting, and successfully obtain exponentially small oscillating terms in the 2-dimensional manifolds., Comment: 6 pages, LaTeX

A proper formulation in the perturbative renormalization group method is presented to deduce amplitude equations. The formulation makes it possible not only avoiding a serious difficulty in the previous reduction to amplitude equations by eliminating all of the secular terms but also consistent derivation of higher-order correction to amplitude equations., Comment: 6 page, revtex

It is shown that the renormalization group method does not necessarily eliminate all secular terms in perturbation series to partial differential equations and a functional subspace of renormalizable secular solutions corresponds to a choice of scales of independent variables in the reductive perturbation method., Comment: 5 pages, latex

Modulational dynamics of oscillatory travelling rolls in magnetoconvection is studied near the onset of a Hopf bifurcation. Using weakly nonlinear analysis, we derive an envelope equation of oscillatory travelling rolls in the plane perpendicular to an ambient vertical magnetic field.The envelope equation is the Davey-Stewartson (DS) equation with complex coefficients, from which we obtain criteria for the modulational (Benjamin-Feir) instability of oscillatory travelling rolls., Comment: 8 pages,latex