Universality and hysteresis in slow sweeping of bifurcations

Bifurcations in dynamical systems are often studied experimentally and numerically using a slow parameter sweep. Focusing on the cases of period-doubling and pitchfork bifurcations in maps, we show that the adiabatic approximation always breaks down sufficiently close to the bifurcation, so the upsweep and downsweep dynamics diverge from one another, disobeying standard bifurcation theory. Nevertheless, we demonstrate universal upsweep and downsweep trajectories for sufficiently slow sweep rates, revealing that the slow trajectories depend essentially on a structural asymmetry parameter, whose effect is negligible for the stationary dynamics. We obtain explicit asymptotic expressions for the universal trajectories and use them to calculate the area of the hysteresis loop enclosed between the upsweep and downsweep trajectories as a function of the asymmetry parameter and the sweep rate.

Figure 1

Instantaneous fixed points (full black lines, stable, and broken black line, unstable) and adiabatic sweep trajectories (gray) of the normal-form family of maps $F_{r,u}$ [defined in (3)], as a function of the parameter $r$, choosing $u=wr, w=0.4$. Instantaneous fixed points are solutions of $F_{r,u}\left(y\right)=y$, showing the standard supercritical pitchfork structure at $r=0$. Adiabatic trajectories are trajectories of (4) with parameters varying according to (5) with $\varepsilon ={10}^{-4}$, and positive and negative sign choice in (5) yielding the upsweep (pale gray) and downsweep (dark gray) trajectory, respectively. Note that since the adiabatic trajectories are shown as a function of the instantaneous value of the parameter $r$ that changes by $\pm \varepsilon$ each time step, the spacing between consecutive points is so small that they overlap in the figure, obscuring the discrete nature of the map dynamics and motivating the continuum approximation, and that time increases from right to left for the downsweep trajectory, as indicated by the arrows. Adiabatic dynamics was calculated starting from a large negative initial time, allowing the trajectory to reach the close vicinity of one of the stable fixed points, and making it independent of the initial value of $x$. We observe that the up- and downsweep trajectories are different, forming a hysteresis loop whose width and height are proportional to ${\varepsilon }^{1/2}$ and ${\varepsilon }^{1/4}$, respectively, up to logarithmic corrections. For $r$ values outside this interval, the adiabatic trajectories follow one of the stable instantaneous fixed points up to small corrections shown in the inset.

Figure 2

Top: Continuum model trajectories from numerical solutions of Eq. (9) in turquoise (thin dashed dark gray) and blue (thin dashed pale gray) for upsweep and downsweep, respectively, overlayed on the trajectories of the map (4) (after transformation from $n$ and $x$ to $t$ and $y/{\varepsilon }^{1/4}$) in pale (upsweep) and dark (downsweep) gray, alongside the instantaneous fixed points of the map in filled (stable) and dashed (unstable) black. Bottom: The difference between the trajectories of the continuum model and the map, showing that as $\varepsilon \to 0$ the trajectories of the continuous- and discrete-time systems approach each other. Here we used $\varepsilon ={10}^{-4},w=0.4$ ($s=0.04$) in the left panels, and $\varepsilon ={10}^{-3},w=0.4$ ($s=0.07$), in the right panels.

Figure 3

Top: The asymptotic approximations of the adiabatic trajectories given in Eqs. (16) and (21) in purple (dark gray) and violet (pale gray) for upsweep and downsweep, respectively, overlayed with the numerically calculated continuum model trajectories of Eq. (9) in turquoise (thin dashed light gray) and blue (thin dashed dark gray) for upsweep and downsweep, respectively. The asymptotic approximations given in Eqs. (16) and (21) are shown each with two curves corresponding to the small-$y$ (left) and large-$y$ (right) regimes, in their overlapping intervals of validity. The breakout time $\tau$, defined in Sec. 3c, is marked in a dotted black line. Bottom: The difference between the asymptotic approximation and the numerically calculated trajectories. In the overlap interval where both the early- and late-time asymptotes are valid, the smaller error in absolute value is shown. As in Fig. 2, $\varepsilon ={10}^{-4},w=0.4$ ($s=0.04$) in the left panels, and $\varepsilon ={10}^{-3},w=0.4$ ($s=0.07$) in the right panels. As expected, the error in the asymptotic approximations decreases with decreasing $\varepsilon$.

Figure 4

The area enclosed within the hysteresis loop in the $t,y$ plane divided by ${\varepsilon }^{1/4}$ versus $ln\left(s\right)$ (natural logarithm of the shape parameter). The area is calculated in four different ways: using the adiabatic trajectories of the normal-form map in gray, the numerically calculated trajectories of the continuum model Eq. (8) in turquoise (light gray), the asymptotic approximations of the continuum trajectories in purple (dark gray), and the explicit asymptotic approximation of the area (25) in black, using $H_0$ as a fit parameter. Circle (respectively, asterisk) symbols are used for cases where the map trajectories were calculated for $w=0.4$ (respectively, $w=0.2$).

Figure 5

Adiabatic sweep trajectories of the shifted and stretched second iterate of the logistic map $N_n$ in black and of the normal map $F_n$ in red (thin dashed gray) for two values of the adiabatic parameter (sweep rate) $\varepsilon$. The agreement is good and improves for decreasing $\varepsilon$.