Origin of chaos near critical points of quantum flow

The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie–Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wave function $\psi$ there is an unstable point (called the $X$ point) in a frame of reference moving with the nodal point. The local phase portrait forms always a characteristic pattern called the “nodal-point–$X$-point complex.” We find general formulas for this complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. We demonstrate that chaos emerges from the consecutive scattering events of the orbits with nodal-point–$X$-point complexes. The scattering events are of two types (called type I and type II). A theoretical model is constructed yielding the local value of the Lyapunov characteristic numbers in scattering events of both types. The local Lyapunov characteristic number scales as an inverse power of the speed of the nodal point in the rest frame, implying that it scales proportionally to the size of the nodal-point–$X$-point complex. It is also an inverse power of the distance of a trajectory from the $X$ point’s stable manifold far from the complex. This distance plays the role of an effective “impact parameter.” The results of detailed numerical experiments with different wave functions, possessing one, two, or three moving nodal points, are reported. Examples are given of regular and chaotic trajectories, and the statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal-point–$X$-point complexes. The numerical results are in agreement with the theory, and various phenomena appearing at first as counterintuitive find a straightforward explanation.

Figure 1

Schematic representation of the nodal-point–$X$-point complex as viewed in a frame of reference of arbitrary velocity $\mathbf{(}{V}_x\left(t\right),V_y\left(t\right))$ at two nearby times $t=t_0$ (solid lines) and $t^{\prime }=t_0+\Delta t$ (dashed lines). $\Delta R_0$ is the length traveled by the nodal point within the time step $\Delta t$. $R_X$ is the distance from the nodal point to the $X$ point at $t=t_0$.

Figure 2

(a) The thick looped curve is the separatrix passing from the $X$ point of the model (17) when ${\dot{x}}_0=3$. The stable and unstable manifolds are marked $S$ and $U$, respectively. The stable manifold crosses the line $u=1$ at $v_s=-0.7785019\dots$ . The upper and lower thin solid lines represent a type-I orbit (initial conditions $u=1$ and $v=v_s+0.01$) and a type-II orbit (initial conditions $u=1$ and $v=v_s-0.005$), respectively. (b) The time growth of the deviations $\xi \left(t\right)$ for the type-I (thick curve) and type-II (thin curve) orbits. In both cases the initial deviation vector is taken ${\vec{\xi }}_0=(1,0)$. The rightmost vertical dashed line at $t=1$ corresponds approximately to the time when the orbits reach $u=-1$—i.e., a position symmetric to their initial conditions with respect to the $v=0$ axis. The two left vertical dashed lines mark the time window $0.45⩽t⩽0.55$ within which the type-I orbit forms part of a loop around the nodal point.

Figure 3

(a) The final value of $\xi ∕{\xi }_0$ (on a logarithmic scale) as a function of the initial condition $v=v_1$ of the orbits of the model (17) taken on the line $u=1$ at $t=0$. The left and right curves correspond to ${\dot{x}}_0=3$ and ${\dot{x}}_0=30$, respectively. The vertical dashed lines mark the position $v=v_s$ at which the $X$ point’s stable manifold $S$ crosses the line $u=1$ in each case. (b) $\xi ∕{\xi }_0$ as a function of $\delta v_1=\mid v_1-v_s\mid$ on a logarithmic scale. The two top curves correspond to ${\dot{x}}_0=3$ (upper curve for type-II orbits, lower curve for type-I orbits). The straight solid lines passing through these curves represent a power-law fitting $\xi ∕{\xi }_0=A\delta v_1^{-b}$ with $b=0.95\dots$ for the upper curve and $b=1.01$ for the lower curve. The bottom two curves correspond to ${\dot{x}}_0=30$. (c) A power-law fitting of the constant $A$ as a function of ${\dot{x}}_0$ [$A=0.89{\dot{x}}_0^{-0.93}$ and $A=2.33{\dot{x}}_0^{-0.89}$ for the lower (type I) and upper (type II) curves, respectively].

Figure 4

The form of the nodal-point–$X$-point complex in the EKC model [44] [equations of motion given by (24) with $a=b=1$, $c=\sqrt{2}∕2$] for the times indicated in panels (a)–(d). The nodal point is an attractor at $t=1.25$ (a). A Hopf bifurcation takes place near $t=1.29415$. The nodal point becomes a repellor, and the associated limit cycle moves outwards at subsequent times. E.g., at $t=1.296$ it has the position shown in (b). At $t=1.303$ the limit cycle reaches the $X$ point (c). Then the limit cycle disappears, and all the orbits inside the nodal-point–$X$-point complex are repelled away from the complex [e.g., at $t=1.35$ (d)].

Figure 5

The value of $⟨f_3⟩$ as a function of the time $t$ in the interval $0⩽t⩽10$ in the EKC model [44] with $a=b=1$ and $c=\sqrt{2}∕2$. The dots mark the points where $⟨f_3⟩=0$.

Figure 6

A type-I event in the EKC model [44] with $a=b=1$ and $c=\sqrt{2}∕2$ and initial conditions of the trajectory, $x_0=1.71510$ and $y_0=1.29285$ at $t=0$. In panels (a)–(e), the nodal point appears as a gray thick dot at (0, 0), while the $X$ point appears as a black thick dot (denoted $X$). The background small arrows indicate the local instantaneous velocity flow in the $(u,v)$ frame of reference. The thick long arrow shows the direction of the deviation vector $\vec{\xi }$ for the same orbit [initial conditions ${\vec{\xi }}_0=(1,0)$]. Panel (f) shows the time evolution of the normalized length $\xi ∕{\xi }_0$ on a logarithmic scale.

Figure 7

Same as in Fig. 6 for a type-II event of the same orbit.

Figure 8

Nodal lines of the wave function $\Psi ={\Psi }_{00}+a{\Psi }_{10}+b{\Psi }_{11}+ϵ{\Psi }_{20}$ when $a=b=1$, $c=\sqrt{2}∕2$, and $ϵ=0.1$.

Figure 9

Nodal lines of the wave function $\Psi ={\Psi }_{00}+a{\Psi }_{20}+b{\Psi }_{11}$ when $c=\sqrt{2}∕2$ and (a) $a=1.23$, $b=1.15$ and (b) $a=1$, $b=1.15$.

Figure 10

(a) Nodal lines of the wave function $\Psi ={\Psi }_{00}+a{\Psi }_{30}+b{\Psi }_{11}$ when $a=1.23$, $b=1.15$, and $c=\sqrt{2}∕2$. (b) The time evolution of $x_0\left(t\right)$ for either one or the three solutions of (44).

Figure 11

Examples of quantum trajectories in the $\psi$ field $\Psi ={\Psi }_{00}+a{\Psi }_{20}+b{\Psi }_{11}$ when $a=1.23$, $b=1.15$, and $c=\sqrt{2}∕2$. (a) An ordered orbit [initial conditions $x\left(0\right)=-1.5$ and $y\left(0\right)=0.1275$], (b) a weakly chaotic orbit [initial conditions $x\left(0\right)=0.850901842117$ and $y\left(0\right)=1.191571712494$], and (c) a strongly chaotic orbit [initial conditions $x\left(0\right)=-1.231356695294$ and $y\left(0\right)=0.840584903955$]. In all three panels the orbits are plotted up to $t=1000$. (d) Time evolution of the “finite-time Lyapunov characteristic number” $\chi \left(t\right)$ for the three orbits.

Figure 12

Histograms (line diagrams) of the number of orbits $\Delta N$ of which the finite time Lyapunov characteristic number at $t={10}^5$ is in the interval $[\chi -0.0025,\chi +0.0025]$, where $\chi$ is the value shown in the abscissa. The orbits result from 500 initial conditions taken randomly in the box $-1.5⩽x⩽1.5$, $0⩽y⩽1.5$. The solid line with squares corresponds to the EKC model [44] (i) $\Psi ={\Psi }_{00}+a{\Psi }_{10}+b{\Psi }_{11}$ (one nodal point), the dashed line to the model (ii) $\Psi ={\Psi }_{00}+a{\Psi }_{20}+b{\Psi }_{11}$ (two nodal points), and the solid line with crossed circles to the model (iii) $\Psi ={\Psi }_{00}+a{\Psi }_{30}+b{\Psi }_{11}$ (three or one nodal point). In all three cases $a=1.23$, $b=1.15$, and $c=\sqrt{2}∕2$. (b) Histogram of the values of ${\log }_{10}{R}_x$ (nodal-point–$X$-point distance) when all the nodal-point–$X$-point complexes are calculated at the times $t=n\times {10}^{-1}$, $n=1,\dots ,{10}^5$. The dotted solid line refers to the EKC model [44] (i), the dashed line to the model (ii), and the solid line to the model (iii), with parameters as in (a).

Figure 13

The finite-time Lyapunov characteristic number $\chi$ at $t={10}^5$ as a function of the number $N(d<0.2)$ of approaches of an orbit to the $X$ point at a distance smaller or equal to 0.2 for the systems (i), (ii), and (iii), shown in (a), (c), and (e), respectively. For each system, 500 initial conditions are taken randomly in the box $-1.5⩽x⩽1.5$, $0⩽y⩽1.5$. The finite-time Lyapunov characteristic number $\chi$ versus the corrected index $N(R_X⩽0.5,d⩽R_X∕2)$ (see text) is shown in (b), (d), and (f), respectively.