We explain the set of rules behind of the LabViewtoolbox for bifurcation analysis of Filippov systems denominated SPTCont 1.0. This software can detect nonsmooth bifurcations in n-dimensional systems using integration-free algorithms based on the evaluation of the vector fields on the discontinuity boundary (DB). In this paper, we present the characteristic point sequences that the software detects to guarantee the existence of local and global nonsmooth bifurcations in planar Filippov systems (n = 2). These sequences can be extended to threedimensional or higher dimension Filippov systems. Boolean-valued functions are used to formulate the conditions of existence for each point defined in the sequences. Dynamics on DB and cycles are defined in function of the set of points. The full catalog of codim 1 local and global bifurcations is used to define the characteristic point sequence when the bifurcation parameter is varied. Finally, an illustrative example is analyzed using step-by-step routines of SPTCont 1.0.
In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS)systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.
La cortadora de tendidos de tela se caracterizada porque el Sistema de Sujeción consiste en que la tela 4 se sujeta mediante un grupo de dos bandas inferiores 13 y dos bandas superiores 14; las bandas superiores 14 tienen un mecanismo para que se desplacen en sentido vertical, disminuya esta distancia y se sujete el tendido de la tela 4; tanto las bandas superiores 14 como las bandas inferiores 13 son movidas por un servomotor 15 dando como resultado que el tendido de tela 4 se desplace en la dirección X con alta precisión y utilizando potencias inferiores a 1.0 KW.