Invariant varieties of discontinuous vector fields. (English) Zbl 1091.34005
The authors focus on a class of so-called standard (reversible) models of small \(C^\infty\)-perturbations of four-dimensional discontinuous vector fields
\[
Z_{a,b,c}(x_1,x_2,x_3,x_4) = L_a(x_1,x_2,x_3,x_4) + P(x_1,x_2,x_3,x_4),
\]
\[ L_a(x_1,x_2,x_3,x_4) = x_2 (\partial /\partial x_1) + x_3 (\partial /\partial x_2) + x_4 (\partial /\partial x_3) + a\; \text{sgn}(x_1) (\partial /\partial x_4), \] where \(P\) is a (given) polynomial and \(a= \pm 1\). Using a particular example first, they provide conditions under which \(Z_{a,b,c}\) present one-parameter families of one-periodic orbits terminating at the origin, provide an exact algorithm to find these orbits, study their local stability and ways in which they disappear, and show how the results can be extended to any small polynomial perturbations of \(L_a\).
\[ L_a(x_1,x_2,x_3,x_4) = x_2 (\partial /\partial x_1) + x_3 (\partial /\partial x_2) + x_4 (\partial /\partial x_3) + a\; \text{sgn}(x_1) (\partial /\partial x_4), \] where \(P\) is a (given) polynomial and \(a= \pm 1\). Using a particular example first, they provide conditions under which \(Z_{a,b,c}\) present one-parameter families of one-periodic orbits terminating at the origin, provide an exact algorithm to find these orbits, study their local stability and ways in which they disappear, and show how the results can be extended to any small polynomial perturbations of \(L_a\).
Reviewer: David S. Boukal (Bergen)
MSC:
34A36 | Discontinuous ordinary differential equations |
34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
34C23 | Bifurcation theory for ordinary differential equations |