Structure of saddle-node and cusp bifurcations of periodic orbits near a non-transversal T-point

Abstract

Non-transversal T-points have been recently found in problems from many different fields: electronic circuits, pendula and laser problems. In this work we study a model, based on the construction of a Poincaré map, that describes the behaviour of curves of saddlenode and cusp bifurcations in the vicinity of such a nontransversal T-point. This model is also able to predict, reproduce and explain the numerical results previously obtained in Chua's equation.

Experimentally Accessible Orbits Near a Bykov Cycle

“…When λ 2 > 0 is increased, the size of the red region diminishes and we glimpse another dominion of [Bykov, 2000] and [Algaba et al, 2010[Algaba et al, , 2011.…”

“…According to [Barrio et al, 2011;Bykov, 2000], for small λ 1 > 0 the line of points (λ 1 , 0) is surrounded by two spiral sheets arbitrarily close to the line within which we observe two families of homoclinic cycles of Shilnikov type, one attracting and another accumulated by horseshoes. These phenomena have also been explored in [Algaba et al, 2011] and [Lamb et al, 2005]; we will give more details about them in Section 10. Among others, references [Aguiar et al, 2005;Algaba et al, 2011;Bykov, 2000;Fernández-Sánchez et al, 2002;Labouriau & Rodrigues, 2012;Rodrigues & Labouriau, 2014;Rodrigues, 2013] have been devoted to the study of T -points and the global bifurcations that they organize in the parameter space.…”

“…These phenomena have also been explored in [Algaba et al, 2011] and [Lamb et al, 2005]; we will give more details about them in Section 10. Among others, references [Aguiar et al, 2005;Algaba et al, 2011;Bykov, 2000;Fernández-Sánchez et al, 2002;Labouriau & Rodrigues, 2012;Rodrigues & Labouriau, 2014;Rodrigues, 2013] have been devoted to the study of T -points and the global bifurcations that they organize in the parameter space. For instance, as the two equilibria are saddle-foci, it is known that two spiral curves, corresponding to homoclinic connections of both equilibria, and a double infinity of spiral curves of saddle-node bifurcations of periodic orbits emerge from each T -point (see Figure 14).…”

“…Taking into account [Barrio et al, 2011[Barrio et al, , 2012Gallas, 2010], this may justify the presence of shrimps and hubs in Figure 2. The mechanism of creation and destruction of dynamics when a third parameter is introduced, as done in [Algaba et al, 2010[Algaba et al, , 2011, is due to the lack of transversality of a curve of T -points in the three-dimensional parameter space.…”

Heterodimensional cycle bifurcation with two orbit flips

“…In the past few decades, the bifurcations of principal homoclinic or heteroclinic orbits in higher dimensional vector fields have been studied extensively (see [1][2][3][4][5][6][7][8][9] and the references cited therein). An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given in [10].…”