Effective equation for two coupled oscillators: Towards a global view of metamorphoses of the amplitude profiles

Localizing Bifurcations in Non-Linear Dynamical Systems via Analytical and Numerical Methods

“…The isolated point indicates the birth of a new branch of solutions, often stable. This is shown in bifurcation diagrams in Figure 9 in [22] or in Figure 4 in [23].…”

“…Such an amplitude curve is shown in Figure 4, and two curves just before and just after metamorphosis are shown in Figure 6 in [22]; see also Figures 5 and 6 in [23]. In this case, an existing branch of stable solution is disrupted; see bifurcation diagrams, Figure 7 in [22] or Figure 7 in [23].…”

“…The corresponding amplitude curves are shown, for example, in Figure 3 in [22] (bluegreen curve) and in Figure 3 in [23] (red curve). The isolated point indicates the birth of a new branch of solutions, often stable.…”

“…It follows that it suffices to demand that Equation ( 16) has multiple roots. Therefore, the formula for the bifurcation set M is [22]:…”

“…We have proposed in our earlier papers an analysis of differential properties of solutions of the amplitude-frequency response Equation (3) [22,23]. It turns out that bifurcations of dynamics, such as hysteresis and jump phenomenon, related to appearance/disappearance of branches of solutions, as well as more complex bifurcations, such as, for example, creation/destruction of solutions, follow from changes of differential properties of solutions of the amplitude-frequency response Equation (3), induced by a change of the parameters c. Note that, from a mathematical point of view, Equation (3) defines an implicit 2D curve.…”