User:Ying-Cheng Lai/Proposed/Transient chaos

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Superpersistent chaotic transients are characterized by the following scaling law for its average lifetime\[\tau \sim \exp{ [C (p - p_c)^{-\chi}] }\ ,\] where \(C > 0\) and \(\chi > 0\) are constants, \(p \ge p_c\) is a bifurcation parameter, and \(p_c\) is its critical value. As \(p\) approaches \(p_c\) from above, the exponent in the exponential dependence diverges, leading to an extremely long transient lifetime. Historically the possibility of such transient raised the question of whether asymptotic attractors are relevant to turbulence.

Superpersistent chaotic transients were first discovered by Grebogi et al. in 1983. In their seminal work, unstable-unstable pair bifurcation was identified as the dynamical mechanism for the transients. In this Review this bifurcation and how it leads to superpersistent chaotic transients will be described. The occurrence of the transients in spatially extended dynamical systems will then be exemplified. Superpersistent chaotic transients associated with the riddling bifurcation that creates a riddled basin of attraction will be discussed, and the effect of noise on the transient lifetimes will be addressed. Finally, application to a physical problem, advection of finite-size particles in open hydrodynamical flows, will be demonstrated.



Chaotic transients in low-dimensional dynamical systems are typically characterized by an algebraic scaling law of its average lifetime \(\tau\) with some parameter variation (Grebogi et al. 1982 and 1983): \[\tag{1} \tau \sim (p - p_c)^{-h}, \ p > p_c, \]

where \(h > 0\) is the algebraic scaling exponent. There exists, however, another distinct class of transient chaos - superpersistent chaotic transients that are characterized by the following scaling law for their average lifetime (Grebogi et al. 1983 and 1985): \[\tag{2} \tau \sim \exp{[C (\Delta p)^{-\chi}]}, \]

where \(\Delta p = p - p_c\ ,\) \(p\) is a system parameter, \(C > 0\) and \(\chi > 0\) are constants. As \(p\) approaches the critical value \(p_c\) from above, the transient lifetime \(\tau\) becomes superpersistent in the sense that the exponent in the exponential dependence diverges. This type of chaotic transients was conceived to occur through the dynamical mechanism of unstable-unstable pair bifurcation, in which an unstable periodic orbit in a chaotic attractor collides with another unstable periodic orbit on the basin boundary (Grebogi et al. 1983 and 1985). The same mechanism causes the riddling bifurcation (Lai et al. 1996) that creates a riddled basin (Alexander et al. 1992), so superpersistent chaotic transients can be expected at the onset of riddling. The transients were also identified in a class of coupled-map lattices, leading to the speculation that asymptotic attractors may not be relevant for turbulence (Crutchfield and Kaneko 1988). Noise-induced superpersistent chaotic transients were demonstrated (Andrade et al. 2000) in phase synchronization (Rosenblum et al. 1996) of weakly coupled chaotic oscillators. Signatures of noise-induced superpersistent chaotic transients were also found (Do and Lai 2003) in the advective dynamics of inertial particles in open fluid flows (Benczik et al. 2002).

Section Unstable-unstable pair bifurcation describes unstable-unstable pair bifurcation and explains why the bifurcation can lead to a superpersistent chaotic transient. Section Riddling bifurcation and superpersistent chaotic transients demonstrates the presence of the chaotic transient at the riddling bifurcation. The next topic is superpersistent chaotic transient in a coupled-map lattice system (Sec. Superpersistent chaotic transients in spatiotemporal systems). The phenomenon of noise-induced superpersistent chaotic transients is described in Sec. Noise-induced superpersistent chaotic transients. An application to advective dynamics of inertial particles in open chaotic flows is presented in Sec. Application: advection of inertial particles in open chaotic flows.

Unstable-unstable pair bifurcation

Unstable-unstable pair bifurcation has been identified as the generic mechanism for superpersistent chaotic transients (Grebogi et al. 1983 and 1985, Lai et al. 1996). One can imagine two unstable periodic orbits of the same periods, one on the chaotic attractor and another on the basin boundary, as shown in Fig. Figure 1(a). In a noiseless situation, as the bifurcation parameter \(p\) reaches a critical value \(p_c\ ,\) the two orbits coalesce and disappear simultaneously, leaving behind a channel in the phase space through which trajectories on the chaotic attractor can escape, as shown in Fig. Figure 1(b). The chaotic attractor is thus converted into a chaotic transient, but the channel created by this mechanism is typically extremely narrow (Grebogi et al. 1983 and 1985, Lai et al. 1996). Suppose on average, it takes time \(T\) for a trajectory to travel through the channel in the phase space so that it is no longer on the attractor, we expect \(T\) to be infinite for \(p = p_c\) but, for \(p > p_c\) the time becomes finite and decreases as \(p\) is increased from \(p_c\ .\) For \(p\) above but close to \(p_c\ ,\) the tunneling time can be long. As we will argue below, we expect \(T\) to increase at least algebraically as \(\Delta p\) is decreased.

From Fig. Figure 1(a), we see that if the phase space is two dimensional, the periodic orbit on the attractor is a saddle and the one on the basin boundary is a repeller. This can arise if the map is noninvertible. Thus, the unstable-unstable pair bifurcation can occur in noninvertible maps of at least dimension two, or in invertible maps of at least dimension three (or in flows of dimension of at least four).

Figure 1: (a) For \(p < p_c\ ,\) a chaotic attractor, the basin boundary, and the pair of unstable periodic orbits. (b) For \(p > p_c\ ,\) escaping channel created by an unstable-unstable pair bifurcation that converts the originally attracting motion into a chaotic transient.

Let \(\lambda > 0\) be the largest Lyapunov exponent of the chaotic attractor. After an unstable-unstable pair bifurcation the opened channel is locally transverse to the attractor. In order for a trajectory to escape, it needs to spend at least time \(T(\Delta p)\) at the location of the opening on the attractor centered about the mediating periodic orbit involved in the bifurcation, stipulating that the trajectory must come to within distance of about \(\exp{[-\lambda T(\Delta p)]}\) from this orbit. The probability for this to occur is proportional to \(\exp[-\lambda T(\Delta p)]\ .\) The average time for the trajectory to remain on the attractor, or the average transient lifetime, is thus \[\tag{3} \tau \sim \exp{ [\lambda T(\Delta p)] }. \]

We see that the dependence of \(T(\Delta p)\) on \(\Delta p\ ,\) which is the average time that trajectories spend in the escaping channel, or the tunneling time, is the key quantity determining the scaling of the average chaotic transient lifetime \(\tau\ .\)

To obtain the scaling dependence of the tunneling time \(T(\Delta p)\) on \(\Delta p\ ,\) we note that, since the escaping channel is extremely narrow, for typical situations where \(\lambda > 0\) and \(T(\Delta p)\) large, the dynamics in the channel is approximately one dimensional along which the periodic orbit on the attractor is stable but the orbit on the basin boundary is unstable for \(p < p_c\) [Fig. Figure 1(a)]. This feature can thus be captured through the following simple one-dimensional map: \[\tag{4} x_{n+1} = x_n^2 + x_n + p, \]

where \(x\) denotes the dynamical variable in the channel and \(p\) is a normalized bifurcation parameter with critical point \(p_c = 0\) [we thus write \(T(p)\)]. For \(p < p_c = 0\ ,\) the map has a stable fixed point \(x_s=-\sqrt{-p}\) and an unstable fixed point \(x_u=\sqrt{-p}\ .\) These two collide at \(p_c\) and disappear for \(p > p_c\ ,\) mimicking an unstable-unstable pair bifurcation.

Since \(T(p)\) is large, map (Eq. (4) can be approximated in continuous-time as \[\tag{5} \frac{dx}{dt} \approx x^2 + p. \]

Suppose the root of the channel is at \(x = 0\) and its length is \(l\ .\) The tunneling time is given by \[\tag{6} T(p) \approx \int^l_0 \frac{dx}{x^2 + p} \sim p^{-1/2}. \]

Substituting Eq. (6) into Eq. (3), we obtain \[\tag{7} \tau (p) \sim \exp{ (C_0 p^{-1/2}) }, \]

where \(C_0 > 0\) is a constant. We see that as \(p\) approaches the critical value \(p_c = 0\) from above, the average transient lifetime diverges in an exponential-algebraic way, giving rise to a superpersistent chaotic transient.

Riddling bifurcation and superpersistent chaotic transients

The presence of symmetry in a dynamical system often leads to an invariant subspace where, in the absence of symmetry-breaking or random perturbations, a trajectory originated in the invariant subspace remains there forever. Situations can also arise where a chaotic attractor lies in the invariant subspace. One common example is the system of coupled, identical chaotic oscillators. The synchronization manifold is naturally a low-dimensional invariant subspace in the full phase space. If another attractor exists outside the invariant subspace, riddling can occur in the sense that the basin of the chaotic attractor in the invariant subspace is riddled with holes of all sizes that belong to the basin of the other attractor. Imagine the situation where all unstable perioic orbits embedded in the chaotic attractor are stable with respect to perturbations in the direction transverse to the invaraint subspace. In this case, almost all initial conditions in the vicinity of the invariant subspace lead to trajectories that end up asymptotically on the chaotic attractor. Riddling bifurcation (Lai et al. 1996) refers to the situation where, when a system parameter changes, an unstable periodic orbit (usually of low period, Hunt and Ott 1996) embedded in the chaotic attractor, becomes transversely unstable. As pointed out in Lai et al. 1996, an immediate physical consequence of the riddling bifurcation is that, when there is a small amount of symmetry-breaking, an extraordinarily low fraction of the trajectories in the invariant subspace diverge. This means that a typical trajectory would spend an extremely long time in the vicinity of the chaotic attractor before approaching the other coexisting attractor. The average lifetime of the chaotic transient versus the amount of symmetry-breaking was shown (Lai et al. 1996) to obey the scaling law for superpersistent chaotic transients.

In a two-dimensional phase space, the invariant subspace is a line. In this case, the onset of riddling is determined by a saddle-repeller bifurcation (eigenvalue +1) (Grebogi et al. 1983 and 1985). A chaotic attractor in the invariant line is typically one-dimensional. Let \(\mathbf{X}_p\) be an unstable fixed point embedded in the chaotic attractor in the invariant subspace. The unstable point is stable transversely to this subspace, as shown in Fig. Figure 2(a). Riddling occurs when \({\mathbf X}_p\) loses its transverse stability as a parameter \(p\) passes through a critical value \(p_c\ .\) The loss of transverse stability is induced by the collision at \(p=p_c\) of two repellers \({\mathbf r}_+\) and \({\mathbf r}_-\ ,\) located symmetrically with respect to the invariant subspace, with the saddle at \({\mathbf x}_p\) (a saddle-repeller pitchfork bifurcation). These two repellers exist only for \(p \leq p_{c}\ ,\) as shown in Fig. Figure 2(a). For \(p>p_c\ ,\) the saddle \({\mathbf x}_p\) becomes a repeller, and the two original repellers \({\mathbf r}_+\) and \({\mathbf r}_-\) off the invariant subspace no longer exist.

Figure 2: (a) Unstable saddle fixed point in the invariant subspace and two repellers off the invariant subspace for \(p<p_c\) (before the saddle-repeller pitchfork bifurcation). (b) Tongue structure formed for \(p > p_c\ ,\) after the onset of riddling. Trajectories originated from initial conditions inside the tongues escape the invariant subspace.

Due to nonlinearity, a tongue opens at \({\mathbf x}_p\ ,\) allowing trajectories near the invariant subspace to escape for \(p> p_c\ ,\) as shown in Fig. Figure 2(b). Each preimage of Figure 2 also developes a tongue simultaneously. Since the preimages of \({\mathbf x}_p\) are dense in the invariant subspace, an infinite number of tongues open simultaneously at \(p=p_c\ ,\) indicating that initial conditions arbitrarily close to the invariant subspace can approach another attractor. Trajectories in the chaotic attractor, however, remain there even for \(p>p_c\ ,\) since the subspace in which the chaotic attractor lies is invariant and each tongue has a zero width there. But trajectories near the chaotic attractor have a finite probability of being in the open and dense set of tongues. The basin of attraction for the chaotic attractor is then a Cantor-set of leaves of positive Lebesgue measure, signifying riddling. Physically, since the onset of riddling induces the supernarrow tongues near the invariant subspace, superpersistent chaotic transient arises (Grebogi et al. 1983 and 1985).

Superpersistent chaotic transients in spatiotemporal systems

An approach to studying spatially extended dynamical system is to examine various spatial patterns and their dynamical evolution. In a turbulent state, the pattern evolution appears random but statistical quantities usually converge for all practical time scales (Frisch 1995). Situations can occur where, after a long time, the system falls onto a low-dimensional attractor. In this case, the high-dimensional, turbulent behavior may be only a transient. It is not possible to determine whether the observed turbulence is transient unless the asymototic time regime is reached. If the transient time is much longer than any physically realizable time, the system is effectively turbulent, regardless of the nature of the asymptotic attractor. In this sense, attractors are not relevant to turbulence. Crutchfield and Kaneko 1988 recognized the possibility of extremely long transient in spatiotemporal dynamical systems. They demonstrated, by using a prototype model, that the attractor can typically be low-dimensional but the transient dynamics can be high-dimensional and complicated. As the system size \(N\) is increased, the transient time can grow exponentially or even faster, as follows: \[\tag{8} \tau_N \sim \exp{(C N^{\alpha})}, \]

where \(C > 0\) and \(\alpha \ge 1\ .\) We see that the transient is superpersistent in the limit \(N \rightarrow\infty\ .\) The case of \(\alpha \le 1\) where the growth of the transient time is exponential or slower with \(N\) was referred to as type-I transient turbulence, while the case \(\alpha > 1\) as type-II transient turbulence (Crutchfield and Kaneko 1988). One example of Type-I transient turbulence is chaotic defect motion in coupled map lattices where the relaxation time for disappearance of the complex patterns increases at most exponentially with the system size. For type-II transient turbulence, the pattern evolution typically appears turbulent and high-dimensional.

To demonstrate superpersistent chaotic transients in spatially extended dynamical systems, Crutchfield and Kaneko 1988 used the following coupled-map-lattice model in which both time and space are discrete but the dynamical variables are continuous: \[\tag{9} x_{n+1}^i = \frac{1}{2r + 1}\sum^{r}_{j = -r}f(x_n^{i+j}), \ \ i = 0, \ldots, N-1, \]

where \(n\) and \(i\) are discrete time and space, respectively, \(f(x)\) is a nonlinear map governing the local dynamics, and \(r\) is a parameter defining the range of spatial coupling. For nearest-neighbor coupling, \(r = 1\ .\) Crutchfield and Kaneko chose the following piecewise linear map, the dripping handrail model, for \(f(x)\ :\) \[\tag{10} f(x) = sx + \omega \ (\mbox{mod} 1), \]

where \(s\) and \(\omega\) are parameters. The local dynamics thus consists of an increase of \(\omega\) with each iteration but when the dynamical variable \(x\) exceeds a threshold \(x_{drop} = (1-\omega)/s\ ,\) a sudden decrease from unity occurs. Physically, the coupled map lattice system Eq. (9) represents a simplified model of a dripping fluid layer, where the local map \(f(x)\) models the dynamics of an isolated drop. The map \(f(x)\) can actually generate complicated dynamics such as chaos and it was also used to study the dynamics of the stirred Belousov-Zhabotinsky chemical reaction (Tsuda 1981).

Crutchfield and Kaneko suggested that both type-I and type-II transient turbulence are due to the complex, hierarchical phase space structure and the transient relaxation can be regarded as a sequence of transitions through a hierarchy of subbasins. These subbasins are subspaces of a basin separated by walls through which a trajectory cannot pass except at portals. In particular, for type-I transient turbulence, the phase space is organized as a hierarchy of subbasins of decreasing dimension. Patterns near the attractor move in relatively low-dimensional subbasins, while those far away from the attractor in high-dimensional subbasins. The collision and annihilation of two defects correspond to an orbit moving from one subbasin to another. Some constant spatial length \(L\) can then be defined for the portal, which is determined by the defect size and the local geometry of the annihilation process. The phase-space volume of the portal is thus \(V \sim c^L\ ,\) where \(c < 1\) is the relative size of the portal with respect to the size of the subbasin. Since the number of defects in a random initial pattern is proportional to \(N\ ,\) the probability for the sequence of transitions down through the hierarchy is \(P_N \sim c^{NL}\ .\) Assuming the dynamics within each subbasin is ergodic, the average transient lifetime is \[\tag{11} \tau_N \sim P_N^{-1} \sim c^{-NL} \equiv (\bar{c})^{NL}, \]

where \(\bar{c} = 1/c > 1\ .\) For type-II transient turbulence, numerical evidence suggests that the patterns are generally complex during the transient epoch but occasionally they can be quite uniform. This implies that the underlying subbasins may consist of long tenrils passing through the neighborhood of the final attractor that corresponds to a simple, uniform pattern. That is, a trajectory can be relatively close to the attractor at some time but most times it moves away from it in order to find the correct path to actually reach the attractor. The subbasin hierarchy can be approximated by a direct product of the local basin structure at each spatial site. The number of subbasins is proportional to \(N^{\gamma}\ ,\) where \(\gamma\) measures the density of the tendrils. Since there are no localized annihilation events, passage through a portal is spatially global. The probability \(p\) of passing a portal is thus \(p \sim c^{N}\ .\) The total probability of passing all portals to reach the final attractor is the product of \(N^{\gamma}\) such local probabilities. The average transient time is \[\tag{12} \tau_N \sim c^{-N^{1+\gamma}} \equiv (\bar{c})^{N^{1 + \gamma}}, \]

which increases faster than exponentially with system size.

This result has been shown to be wrong in a successive study by Politi, Livi, Oppo, Kapral (1993) in that paper the authors have shown, by performing accurate numerical simulations, that the transients increase as simple exponential with the system size and not super-exponentially as reported by Crutchfield and Kaneko (1988). Moreover, the mechanisms underlying such a behaviour was identified and termed Stable Chaos, since exponential transients can occur in spatially extended systems even for non chaotic maps and are due to finite amplitude effects (for more details see T. Tel and Y-C Lai 2008). This field has led to a fruitful research direction, in the last 20 years, which has found applications in field as different as chaotic synchronization and dynamics of neural networks (for a recent review see Politi and Torcini, 2010).

Noise-induced superpersistent chaotic transients

In the general setting where an unstable-unstable pair bifurcation occurs, noise can induce superpersistent chaotic transients preceding the bifurcation. Consider, in the noiseless case, a chaotic attractor and its basin of attraction. When noise is present, there can be a nonzero probability that two periodic orbits, one belonging to the attractor and another to the basin boundary, get close and coalesce temporally, giving rise to a nonzero probability that a trajectory on the chaotic attractor crosses the basin boundary and moves to the basin of another attractor. Transient chaos thus arises. Due to noise, the channels through which trajectory escapes the chaotic attractor open and close intermittently in time. The probability of escape is extremely small because escaping through the channel requires staying of the trajectory in a small vicinity of the opening of the channel consecutively for a finite amount of time, which is an event with extremely small probability. In this sense, the channel must be super narrow (Grebogi et al. 1983 and 1985, Lai et al. 1996), leading to a superpersistent chaotic transient. The creation of the channel by noise and the noisy dynamics in the channel are thus the key to understanding the noise-induced transient behavior.

There are two regimes of interest. In the subcritical case, there is a chaotic attractor and no escaping channel exists in the absence of noise. In this case, the channel is induced by noise and it opens and closes randomly in time. In the supercritical case, the channel is open and there is already a superpersistent chaotic transient. The presence of noise affects the deterministic dynamics in the channel. In both cases, the dynamics in the channel can be regarded as being driven by a stochastic force and, hence, it can be modeled by a stochastic differential equation, the solution to which gives the tunneling time through the channel. Apparently, this time depends on the noise amplitude. The dependence, in combination with the small probability for a trajectory to move to the opening of the channel and to stay there for the duration of the tunneling time, gives the scaling of the average lifetime of the superpersistent chaotic transients with the noise amplitude.

Let \(\varepsilon\) be the noise amplitude. To obtain the scaling dependence of the tunneling time \(T(\varepsilon)\) on \(\varepsilon\ ,\) we use the following one-dimensional map: \[\tag{13} x_{n+1} = x_n^{k-1} + x_n + p + \varepsilon {\mathbf \xi}(n), \]

where \(k \ge 3\) is an odd integer so as to generate a pair of fixed points with different unstable dimension, \(p_c = 0\ ,\) and \(\xi (n)\) is a Gaussian random process of zero mean and unit variance. If the tunneling time is \(T \gg 1\ ,\) Eq. (13) can be approximated by \[\tag{14} \frac{dx}{dt} = x^{k-1} + p +\varepsilon{\mathbf \xi}(t), \]

For \(p<0\ ,\) the deterministic system for Eq. (14) has a stable fixed point \(x_s=-|p|^{1/(k-1)}\) and an unstable fixed point \(x_u=|p|^{1/(k-1)}\ ,\) but there are no more fixed points for \(p>0\ .\) Let \(x_r=x_s\) for \(p<0\) and \(x_r=0\) for \(p\ge 0\ ,\) and let \(T_p^k\) be the tunneling time. A properly formulated first-passage-time problem for this one-dimensional stochastic process yields the scaling of \(T_p^k\) with the noise amplitude \(\varepsilon\) (Do and Lai 2004 and 2005).

Let \(P(x,t)\) be a probability density function of the stochastic process governed by Eq. (14), which satisfies the Fokker-Planck equation \cite{Gardiner:book,Risken:book}: \[\tag{15} \frac{\partial P(x,t)}{\partial t} = -\frac{\partial}{\partial x}[(x^{k-1} + p) P(x,t)] +\frac{\varepsilon^2}{2}\frac{\partial^2 P}{\partial x^2}. \]

Let \(l\) be the effective length of the channel in the sense that a trajectory with \(x > l\) is considered to have escaped the channel. The time required for a trajectory to travel through the channel is equivalent to the mean first passage time T from \(x_r\) to \(l\ .\) Focusing on a trajectory that escapes eventually, we assume that, once it falls into the channel through \(x_r\ ,\) it will eventually exit the channel at \(x = l\) without returning to the original chaotic attractor. This is reasonable considering that the probability for a trajectory to fall in the channel and then to escape is already exponentially small and, hence, the probability for any second-order process to occur, where a trajectory falls in the channel, moves back to the original attractor, and falls back in the channel again, is negligible. For trajectories in the channel there is thus a reflecting boundary condition at \(x = 0\ :\) \[\tag{16} [P(x,t) - \frac{\partial P}{\partial x}]|_{x = x_r} = 0. \]

That trajectories exit the channel at \(x = l\) indicates an absorbing boundary condition at \(x = l\ :\) \[\tag{17} P(l,t) = 0. \]

Assuming that trajectories initially are near the opening of the channel (but in the channel), we have the initial condition \[\tag{18} P(x,x_r) = \delta (x - x_r^+). \]

Under these boundary and initial conditions, the solution to the Fokker-Planck equation yields the following mean first-passage-time (Gardiner 1997, Risken 1989) for the stochastic process (Eq. (14): \[\tag{19} T_{p}^k(\varepsilon) = \frac{2}{\varepsilon^2} \int_{x_r}^l dy \exp{[-b H(y)]} \int^y_{x_r} \exp{[b H(y')]} dy' \]

where \(H(x)=(x^k + kpx)\) and \(b=2/(k\varepsilon^2)\ .\)

The double integral in Eq. (14) can be carried out (Do and Lai 2004 and 2005) for the three distinct cases: critical (\(p = 0\)), supercritical (\(p > 0\)), and subcritical (\(p < 0\)). The results can be summarized as follows.

  • For the small noise regime (\(\varepsilon \ll \varepsilon_c \sim |p|^{k/(2(k-1))}\)),

\[\tag{20} T_p^k(\varepsilon) \sim \begin{cases} p^{-(k-2)/(k-1)}, & p>0,\\ \varepsilon^{-(2-4/k)}, & p=0, \\ |p|^{-(k-2)/(k-1)}\exp(\frac{|p|^{k/(k-1)}}{\varepsilon^2}), & p<0. \end{cases} \]

  • For the large noise regime (\(\varepsilon \gg \varepsilon_c\)),

\[\tag{21} T_p^k(\varepsilon) \sim \varepsilon^{-(2-4/k)}. \]

These laws imply the following scaling laws for the average lifetime of the chaotic transients in various regimes:

  • For the small noise regime (\(\varepsilon \ll \varepsilon_c \sim |p|^{k/(2(k-1))}\)),

\[\tag{22} \tau_p^k(\varepsilon) \sim \begin{cases} \exp{[ p^{-(k-2)/(k-1)} ]}, & p>0,\\ \exp{[\varepsilon^{-(2-4/k)} ]}, & p=0, \\ \exp{ (|p|^{-(k-2)/(k-1)}\exp[{|p|^{k/(k-1)}}/{\varepsilon^2}])}, & p<0. \end{cases} \]

  • For the large noise regime (\(\varepsilon \gg \varepsilon_c\)), we have

\[\tag{23} \tau_p^k(\varepsilon) \sim \exp( \varepsilon^{-(2-4/k)}). \]

The general observation is that for large noise (\(\varepsilon \gg \varepsilon_c\)), the transient is normally superpersistent. For small noise, three behaviors arise depending on the bifurcation parameter \(p\ :\) constant (independent of noise) for the supercritical regime, normally superpersistent for the critical case, and extraordinarily superpersistent for the subcritical regime in the sense of scaling in Eq. (22) (for \(p < 0\)). Numerical support for these distinct scaling behaviors can be found in Do and Lai 2004 and 2005.

Application: advection of inertial particles in open chaotic flows

The phenomenon of superpersistent chaotic transients finds application in fluid dynamics. It has been known that ideal particles of zero mass and size follow the velocity of the flow and, as such, the advective dynamics can be described as Hamiltonian (Aref 1984, Ottino 1989) in the physical space for which chaos can arise but not attractors. In an open Hamiltonian flow, ideal particles coming from the upper stream must necessarily go out of the region of interest in finite time. However, the inertia of the advective particles can alter the flow locally (Maxey and Riley 1983). As a result, the underlying dynamical system becomes dissipative for which attractors can arise and, hence, particles can be trapped permanently in some region in the physical space (Rubin et al. 1995, Burns et al. 1999). This phenomenon was demonstrated in a model of two-dimensional flow past a cylindrical obstacle (Benczik et al. 2002). This result has implications in environmental science where forecasting aerosol and pollutant transport is a basic task, or even in defense where the spill of a toxin or biological pathogen in large-scale flows is of critical concern. The possibility that toxin particles can be trapped in physical space is particularly worrisome. It is thus interesting to study the structural stability of such attractors (Do and Lai 2003). In particular, can chaotic attractors so formed be persistent under small noise? It was found (Do and Lai 2003) that in general, the attractor is destroyed by small noise and replaced by a chaotic transient, which is typically superpersistent. For small noise, the extraordinarily long trapping time makes the transient particle motion practically equivalent to an attracting motion with similar physical or biological effects. This finding suggests a way to directly observe superpersistent chaotic transients in laboratory experiments.

For an ideal, passive particle of zero inertia and zero size advected in a flow, the particle velocity \({\mathbf v}\) is the flow velocity \({\mathbf u}\) which, in a two-dimensional physical space, is determined by a stream function \(\Psi (x,y,t)\ :\) \(u_x = \partial \Psi/\partial y\) and \(u_y = - \partial \Psi/\partial x\ .\) For particles of finite size, viscous friction arises and, as such, their velocities differ from those of the fluid. Consider a spherical particle of radius \(a\) and mass \(m_p\ ,\) and fluid of dynamic viscosity \(\mu\) and element mass \(m_f\ ,\) the equation of motion of the advective particle is (Maxey and Riley 1983) \[ m_p \frac{d{\mathbf v}}{dt} = m_f \frac{d{\mathbf u}}{dt} - (m_f/2)(\frac{d{\mathbf v}}{dt} - \frac{d{\mathbf u}}{dt}) - 6\pi a\mu ({\mathbf v} - {\mathbf u}), \] where on the right-hand side, the first term is the fluid force from the undisturbed flow field, the second term is the force due to the added mass effect, and the third represents the Stokes drag. While in principle, the fluid velocity \({\mathbf u}\) is disturbed by the particle motion, if the particle sizes are relatively small and their concentration is low, \({\mathbf u}\) can be considered as unchanged (Benczik et al. 2002). For convenience, one can introduce the mass ratio parameter \[ R = \frac{2\rho_f}{\rho_f + 2\rho_p} \] and the inertial parameter \[ A = \frac{R}{\frac{2}{9}(a/L)^2 R_e}, \] where \(\rho_p\) and \(\rho_f\) are the densities of the particle and of the fluid, respectively, \(L\) is a typical large-scale mixing length, and \(R_e\) is the Reynolds number. The equation of motion can then be casted into a dimensionless form. To simulate random forcing due to flow disturbance or other environmental factors, we add terms \(\varepsilon\xi_x(t)\) and \(\varepsilon\xi_y(t)\) to the force components in the \(x\)- and \(y\)-directions, where \(\xi_x(t)\) and \(\xi_y(t)\) are independent Gaussian random variables of zero mean and unit variance, and \(\varepsilon\) is the noise amplitude. The final equation of motion under random perturbations is \[ \frac{d{\mathbf v}}{dt} - \frac{3R}{2} \frac{d{\mathbf u}}{dt} = - A ({\mathbf v} - {\mathbf u}) + \varepsilon {\mathbf \xi}(t), \] where \({\mathbf \xi}(t) = [\xi_x(t), \xi_y(t)]^T\ .\) Inertial particles are aerosols if \(0 < R < 2/3\) and they are bubbles if \(2/3 < R < 2\ .\) The limit \(A \rightarrow\infty\) corresponds to the situation of ideal particles (passive advection).

A convenient numerical framework to study the advective dynamics of inertial particles (Benczik et al. 2002) is the open flow model of the von Kármán vortex street in the wake of a cylinder of radius \(r\ ,\) located at \((x,y) = (0,0)\ .\) A time-periodic stream function \(\Psi(x,y,t)\) (period \(T_f = 1\) in a standard dimensionless form) governing the motions of vortices in the background flow of velocity \(u_0\) can be constructed explicitly from the solutions of the two-dimensional viscous Navier-Stokes equations for the geometry of a circle of radius \(r\) in the middle of an infinite channel of width \(w = 4r\) (Jung et al. 1993). The Reynolds number is \(R_e \approx 250\ .\) The flow velocity \({\mathbf u}(x,y,t)\) can be obtained from \(\Psi(x,y,t)\ ,\) allowing the particle motions to be computed.

In Benczik et al. 2002, it was shown that attractors can be formed in the bubble regime. It is thus convenient to focus on this regime, e.g., by fixing \(R = 1.47\) and \(A = 30\ .\) There are three attractors (Benczik et al. 2002): two chaotic and one at \(x = \infty\ .\) The chaotic attractors are located near the cylinder (but not stuck on it): one in \(y > 0\) and another in \(y < 0\ .\) To gain insight into what might happen to the attractors under noise, the basins of attraction of these attractors can be examined (Do and Lai 2003). Near the cylinder, the basin boundaries among the three attractors are apparently fractal. Because of the explicit time dependence in the stream function and therefore in the flow velocities, the attractors and their basins move oscillatorily around the cylinder. The remarkable feature is that in the physical space, there are time intervals during which the attractors come close to the basin boundaries. Thus, under noise, we expect permanently trapped motion on any one of the two chaotic attractors to become impossible. In particular, particles can be trapped near the cylinder, switching intermittently between the two originally chaotic attractors, but this can last only for a finite amount of time: eventually all trajectories on these attractors escape and approach the \(x = \infty\) attractor. That is, chaos becomes transient under noise.

To understand the nature of the noise-induced transient chaos, one can distribute a large number of particles in the original basins of the chaotic attractors and examine the channel(s) through which they escape to the \(x = \infty\) attractor under noise (Do and Lai 2003). Due to the symmetry of the flow (Jung et al. 1993), the particle trajectories at \(t\) and \(t + T_f/2\) are symmetric to each other with respect to the \(x\)-axis. While there are particles still trapped in the original attractors, many others are already away from the cylinder. The channels through which they escape are a set of thin openings surrounding the cylinder and extending to one of the vortices in the flow. After wandering near the vortex, particles go to the \(x = \infty\) attractor. Because of the time-dependent nature of the flow, in the physical space the locations of these openings vary in time, but the feature that they are narrow is common. For a fixed noise amplitude, numerically it was found that the lifetimes of the particles near the cylinder obey an extremely slow, exponentially decaying distribution. The scaling law between the average transient lifetime and the noise amplitude turns out to be characteristically of that of a superpersistent chaotic transient (Do and Lai 2003). Theoretically, the observed noise-induced superpersistent chaotic transient can be explained by using the approach in Sec. Noise-induced superpersistent chaotic transients. Since the phase space in a two-dimensional fluid problem is exactly the configuration (or physical) space, the result implies that it may be possible to observe superpersistent chaotic transients in physical space. It was suggested (Do and Lai 2003) that the flow system used for experimental study of chaotic scattering (Sommerer et al. 1996) could be used for this purpose.


In conclusion, an unstable-unstable pair bifurcation can generate a narrow channel through which trajectories originally on a chaotic attractor can escape, converting attracting motion into a transient. The average transient lifetime depends exponentially on the time required for a trajectory to pass the channel, which in turn depends on quantities such as the parameter difference, symmetry-breaking parameter, and noise amplitude etc., typically algebraically. As a result, a superpersistent chaotic transient arises. The transients can accompany phenomenon such as the onset of riddled basins and the stability of attractors formed by inertial particles advected in hydrodynamical fluid flows. Such transients are also expected to be common in spatially extended dynamical systems (For a recent review on this topic, see Tél and Lai 2008).


This work was supported by AFOSR under Grant No. FA9550-06-1-0024.


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Thiel, M.; Kurths, J.; Romano, M.C.; Moura, A.; Karolyi, G., (Springer Verlag, 2010, Heidelberg)

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