Duffing-type Equations: Singular Points of Amplitude Profiles and Bifurcations

Abstract

We reemphasize that the ratio R sµ ≡B(B s → µμ)/∆M s is a measure of the tension of the Standard Model (SM) with the latest measurements ofB(B s → µμ) that does not suffer from the persistent puzzle on the |V cb | determinations from inclusive versus exclusive b → c ν decays and which affects the value of the CKM element |V ts | that is crucial for the SM predictions of bothB(B s → µμ) and ∆M s , but cancels out in the ratio R sµ . In our analysis, we include higher-order electroweak and QED corrections and adapt the latest hadronic input to find a tension of about 2σ for R sµ measurements with the SM independently of |V ts |. We also discuss the ratio R dµ which could turn out, in particular in correlation with R sµ , to be useful for the search for new physics, when data on both ratios improve. R dµ is also independent of |V cb | or more precisely |V td |.

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

“…= 0 [23] (this was first noticed for the Duffing equation in [24]), while more complicated qualitative changes of these curves follow from the conditions (defining a singular point of the curve)…”

“…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].…”

“…In Section 2 we describe our methodology and classify a wide variety of metamorphoses; then, in Section 3, we show applications of our approach to two dynamical systems. Firstly, we consider a damped periodically driven pendulum, applying our results obtained in [23]. Secondly, we study a driven pendulum with van der Pol's type damping, generalizing our previous studies [30,31].…”

“…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.…”

“…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.…”