Analysis of a slow–fast system near a cusp singularity

Abstract

International audienceThis paper studies a slow fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe. Our analysis consists of two main parts: first, we recall a formal normal form suitable for systems as the one studied here; afterwards, taking advantage of this normal form, we investigate the transition near the cusp singularity by means of the blow up technique. Our contribution relies heavily in the usage of normal form theory, allowing us to refine previous results. (C) 2015 Elsevier Inc. All rights reserved

Stabilization of a class of slow–fast control systems at non-hyperbolic points

“…In the context of slow-fast systems it was first introduced in [4], and has been further developed afterwards, see e.g. [10,11,[19][20][21] and [12][13][14][15] for some applications in control systems.…”

“…. ,r 2x k−1 ,rz,r γ ), (11) where γ∈ N shall be appropriately chosen below. Next, we obtain the blown up vector field.…”

Formal normal form of Ak slow–fast systems

“…Theorem 2.2, together with Borel's lemma [3], implies that an A k -SFS X = F + P is smoothly conjugate to a smooth vector field Y = F + H where H is flat at the origin. The benefits of this normal form are exploited in [6,7].…”

“…In this case, the blow-up technique [4,9] can be applied to desingularize the SFS. This methodology has been successfully used in many cases, e.g., [2,7,10,11,15,16], where many of these deal with an A k -SFS with fixed k = 2 or k = 3. Briefly speaking, the blow-up technique consists in an appropriate change of coordinates under which the induced vector field is regular or has simpler singularities (hyperbolic or partially hyperbolic).…”

“…As it is shown below, the structure of the A k -SFS plays an important role in the normalization process. Moreover, this normalization greatly simplifies the local analysis of systems given by (4), as shown in [6,7].…”