Dynamical Systems I

Normal forms of planar switching systems

“…Further, we prove that only one of those 19 types of linear normal form decides system (1) to be monodromic ( [1]), i.e., no orbits approaching O in a definite direction as t → ±∞. In section 3 we show our normalization with second order terms and normalized linear terms, constructing a near-identity transformation with switching to eliminate some second order terms for the second order normal form of system (1). In section 4 we show how the monodromic linear normal form can be used to simplify the above obtained second order normal form further and give the second order monodromic normal form.…”

“…All orbit structures are persistent in normal forms including the possible sliding motions because of topological equivalence. Further, we prove that only one of those 19 types of linear normal form decides system (1) to be monodromic ( [1]), i.e., no orbits approaching O in a definite direction as t → ±∞. In section 3 we show our normalization with second order terms and normalized linear terms, constructing a near-identity transformation with switching to eliminate some second order terms for the second order normal form of system (1).…”

“…Normalization of linear terms. We need to simplify the linear part at first, i.e., to give a 'linear normal form' for system (1). For this purpose we need to simplify the two matrices A ± on the two sides of the switching line y = 0 by a continuous and invertible piecewise-linear transformation…”

“…It shows that O is monodromic of FF type ( [11]), i.e., system (1) is monodromic at O only in the case that both the upper system and the lower one are monodromic, i.e., A ± have a pair of complex eigenvalues separately. The monodromic O can be of neither PP type nor FP type since O is assumed to be an equilibrium of both the upper system and the lower one in the beginning of this paper (just before (1). Note in Corollary 2 that orbits near O rotate clockwise (resp.…”