Studies on Dynamical systems

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A new formalism for gauge connections is given and monlinear gauge eauations are solved and the divergence of quantu field theory is discussed. The following results are obtained.(1) A gauge theory is formulated in termas of a certain kind of a decomposition of a linear space, whitch is called a "gauge decomposition." Then every gauge connection connection can be extended to a flat connection, when it admits a representation in some algebra (flat extension theorem).(2) By use of the result (1), nonlinear gauge equations can be solved by use of a decomposition of a solution of a linear equation. Symmetry structures and special class of silutions can be discussed as a "Galois thery for differntial equations". When it has a representation in "algebroids", divergece appears and anomaly and renormalization can discussed in this case.