SLOW PASSAGE THROUGH MULTIPLE BIFURCATION POINTS

The slow passage problem, the slow variation of a control parameter, 
is explored in a model problem that posses several co-existing 
equilibria (fixed points, limit cycles and 2-tori), and these are 
either created or destroyed or change their stability as control 
parameters are varied through Hopf, Neimark-Sacker and torus break-up 
bifurcations. The slow passage through the Hopf bifurcation behaves as 
determined in previous studies (the delay in the observation of 
oscillations depends only on how far from critical the ramped 
parameter is at the start of the ramp--a memory effect), and that 
through the Neimark-Sacker bifurcation also behaves similarly. We show 
that the range of the ramped parameter over which a Hopf oscillation 
can be observed (limited by the subsequent onset of torus oscillations 
from the Neimark-Sacker bifurcation) is twice that predicted from a 
static-parameter bifurcation analysis, and this is a memory-less 
result independent of the initial value of the ramped parameter. 
These delay and memory effects are independent of the ramp rate, for 
small enough ramp rates. The slow passage through the torus break-up 
bifurcation is qualitatively different. It does not depend on the 
initial value of the ramped parameter, but instead is found to depend, 
on average, on the square-root of the ramp rate. This is typical of 
transient behavior. We show that this transient behavior is due to the 
stable and unstable manifolds of the saddle limit cycles forming a 
very narrow escape tunnel for trajectories originating near the 
unstable 2-torus no matter how slow a ramp speed is used. The type of 
bifurcation sequence in the model problem studied (Hopf, 
Neimark-Sacker, torus break-up) is typical of those for the transition 
to spatio-temporal chaos in hydrodynamic problems, and in those 
physical problems the transition can occur over a very small range of 
the control parameter, and so the inevitable slow drift of the 
parameter in an experiment may lead to observations where the slow 
passage results reported here need to be taken into account.


(Communicated by Yang Kuang)
Abstract.The slow passage problem, the slow variation of a control parameter, is explored in a model problem that posses several co-existing equilibria (fixed points, limit cycles and 2-tori), and these are either created or destroyed or change their stability as control parameters are varied through Hopf, Neimark-Sacker and torus break-up bifurcations.The slow passage through the Hopf bifurcation behaves as determined in previous studies (the delay in the observation of oscillations depends only on how far from critical the ramped parameter is at the start of the ramp-a memory effect), and that through the Neimark-Sacker bifurcation also behaves similarly.We show that the range of the ramped parameter over which a Hopf oscillation can be observed (limited by the subsequent onset of torus oscillations from the Neimark-Sacker bifurcation) is twice that predicted from a static-parameter bifurcation analysis, and this is a memory-less result independent of the initial value of the ramped parameter.These delay and memory effects are independent of the ramp rate, for small enough ramp rates.The slow passage through the torus break-up bifurcation is qualitatively different.It does not depend on the initial value of the ramped parameter, but instead is found to depend, on average, on the squareroot of the ramp rate.This is typical of transient behavior.We show that this transient behavior is due to the stable and unstable manifolds of the saddle limit cycles forming a very narrow escape tunnel for trajectories originating near the unstable 2-torus no matter how slow a ramp speed is used.The type of bifurcation sequence in the model problem studied (Hopf, Neimark-Sacker, torus break-up) is typical of those for the transition to spatio-temporal chaos in hydrodynamic problems, and in those physical problems the transition can occur over a very small range of the control parameter, and so the inevitable slow drift of the parameter in an experiment may lead to observations where the slow passage results reported here need to be taken into account.
1. Introduction.There has been much interest in systems with slowly varying control parameters.Such problems naturally arise as models of systems with multiple scales.The desire to understand certain complicated multi-scale dynamics by 96 YOUNGHAE DO AND JUAN M. LOPEZ studying the bifurcation structure of the fast processes when the slow processes are treated as slowly varying control parameters motivated the early work on the slow passage problem [1].For example, inter-cellular calcium oscillations have slow-fast characteristics with the ratio of the slow to fast time-scales being experimentally observed to be in the range of 10 −1 to 10 −2 [2]; slow passage effects near bifurcation instabilities of these systems have recently been studied [3].Also, the control parameters in certain systems may inherently vary on a time scale much slower than the time scale of the dynamics, such as in chemical reactions where the reactant concentrations vary slowly on the intrinsic time scale of the reactors.Such systems have also been studied in the context of the slow passage problem [4].The slow passage effect is that transition may not occur until the parameter is considerably beyond the critical value predicted from a static bifurcation analysis (delay effect), and that the delay in onset is dependent on the initial state of the system (memory effect).Recently, a new effect for slow passage, called an early effect has been found [5].In the early effect scenario, transition may occur well before the parameter reaches the value predicted from a static bifurcation analysis.
In other experimental systems, such as in fluid dynamics, the experimental protocol sometimes uses a slow variation of the control parameter in order to explore the nonlinear dynamics of the system over a range of the control parameter in a reasonable time [6,7,8,9].In utilizing such protocols, it is often argued that if the time variation of the control parameter is slow enough, the system adjusts adiabatically and that in such a quasi-static regime one recovers the bifurcation dynamics of the system (i.e., that the quasi-static state at the instantaneous value of the control parameter is the same as the state at the fixed value of the parameter after transients have decayed).However, model problems considering the slow variation of a control parameter through a Hopf bifurcation have found that the delay effect is independent of the rate of variation of the parameter, especially in the limit of slow variations.This raises concerns about so-called quasi-static parameter variations.Conducting a theoretical or numerical analysis of the slow passage problem for some systems of interest, such as fully three-dimensional fluid dynamics, is prohibitively expensive.One conceivably could try performing a center manifold/normal form reduction in the neighborhood of an instability and then perform a slow passage analysis, but the validity of the reduced system would still be in question.This type of analysis has been conducted, e.g.[10,11], where the slow passage problem has been considered in the neighborhood of a Neimark-Sacker bifurcation.In a practical setting however, the variation of the parameter is typically over an extensive range well beyond the neighborhood in which the normal form is valid, and may cover a number of other bifurcations.
The slow passage problem past a Hopf bifurcation has been widely studied [1,12,4,13].The slow passage problem through several other bifurcations have also been studied, such as saddle-node bifurcations and through resonances [14,15,16].Those studies have all considered very low-dimensional model problems.The analysis of ODE systems of degree greater than two has also been generalized [17], but very little (none that we are aware of) has been done on the slow passage problem on secondary instabilities.In many problems of interest, particularly in experiments where a quasi-static variation of a control parameter is used to investigate a sequence of instabilities a system undergoes over a range of the control parameter, an understanding of what to expect as the parameter is varied across secondary instabilities is lacking.Hall [18] has raised related concerns about the use of the quasi-steady approach in situations where multiple solutions associated with nearly coincident eigenvalues occur.At some values of the appropriate stability parameter, more than one stable configuration is possible.Hall noted that the ultimate (if any) stable state then depends on the form of the initial perturbation, and it can be different from that predicted using a quasi-steady approach.
In this paper, we consider a slow passage problem in a model problem with multiple dynamical states and bifurcation points.It is an ODE system of small dimension which has been previously analyzed using a bifurcation-theoretic approach and shown to posses several co-existing equilibria (fixed points, limit cycles and 2-tori), and these may be created, destroyed or change their stability as control parameters are varied through Hopf, Neimark-Sacker and torus break-up bifurcations [19].The model equation is derived from a complicated double pendulum system with two forcing terms, three springs and two control parameters ν and µ.Here, we fix ν and treat µ as a slowly varying parameter to explore the slow passage problem through the rich dynamical landscape.
The model equation we consider is very similar to a normal form of a double Hopf bifurcation [20].The double Hopf bifurcation admits several scenarios distinguished by the values of the normal form coefficients.The double Hopf bifurcation associated with the model problem under consideration is a degenerate version of case 6 of the difficult scenarios, as classified by Kuznetsov [20].Double Hopf bifurcations are of broad interest, and in fluid dynamics problems their occurrence is quite common [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35].
In Sec. 2 we briefly summarize the model and its static bifurcation dynamics (summarizing the results of [19]).Section 3 investigates the nonlinear dynamics as µ is slowly increased with a linear ramp through the various bifurcations.Based on our analytical and numerical results for the onset conditions, we show that the slow passage through the Hopf bifurcation behaves as determined in previous studies, and that the slow passage through the Neimark-Sacker bifurcation behaves similarly.The delay effect depends only on how far from critical the ramped parameter is at the start of the ramp and is independent of the ramp rate (for slow enough ramps; in this model problem we give specific bounds on how slow the ramp must be).In contrast, the slow passage through the torus break-up bifurcation does not depend on the initial value of the ramped parameter, but instead is found to depend, on average, on the square-root of the ramp rate.This is typical of transient behavior.We show that this transient behavior is due to the stable and unstable manifolds of the saddle limit cycles involved forming a very narrow escape tunnel for trajectories originating near the unstable 2-torus.Finally, the results are summarized in Sec. 4.

Model equations.
To study the slow passage problem in a nonlinear lowdimensional problem with Hopf and Neimark-Sacker and torus break-up bifurcations, we consider the following dynamical system where r i and θ i are amplitude and phase variables, respectively, and the control parameter µ as a slow linear function of time, given by When the ramp rate = 0, µ is a static bifurcation parameter.In this section, we summarize the bifurcation analysis of the nonlinear dynamics presented in [19], as the two control parameters ν and µ are varied, with = 0. Note that the moduli equations ( 1)-( 2) decouple from the phase equations ( 3)-( 4).Considering the ṙ1 and ṙ2 equations first, they have four equilibria: The equilibria have the following existence conditions, ν > 0 and: 1. P 1 is a fixed point and exists for all ν and µ; 2. P 2 is a limit cycle and exists if µ ≥ 88ν/75; 3. P 3 is a limit cycle and exists if µ ≥ 74ν/159; 4. P 4 is a 2-torus and exists if µ ≥ 7478ν/3975.The stability of the equilibria can be determined from the eigenvalues of the Jacobian matrix evaluated at the equilibrium.For P 1 the eigenvalues are λ 1,2 = −11ν/7 + 75µ/56 and 37ν/28 − 159µ/56, for P 2 the eigenvalues are λ 1,2 = 22ν/7 − 75µ/28 and −3739ν/700 + 159µ/56, and for P 3 the eigenvalues are λ 1,2 = −241ν/742 − 75µ/56 and −37ν/14 + 159µ/28.In the slow passage analysis, we shall fix ν = 0.5.For P 4 with ν = 0.5, if µ NS ≤ µ ≤ µ T ≈ 1.88, the eigenvalues λ 1,2 = a ± bi are complex conjugates with a ≤ 0, and if µ ≥ µ T the real parts of the eigenvalues are positive.
Bifurcation diagrams for ν = 0.5 are given in Fig. 1.For ν = 0.5, the fixed point P 1 changes stability from a saddle (unstable), with r 2 its unstable manifold and r 1 its stable manifold, to a stable node at a Hopf bifurcation at µ = µ H1 = 37/159, spawning the unstable limit cycle P 3 with r 1 = 0.At µ = µ H2 = 44/75, P 1 changes from stable to unstable at a second Hopf bifurcation spawning the stable limit cycle P 2 with r 2 = 0. P 2 loses stability in a non-resonant Neimark-Sacker bifurcation at µ = µ NS = 3739/3975, spawning the 2-torus P 4 with r 1 = 0 and r 2 = 0.At µ = µ T ≈ 1.88, P 4 loses stability in a torus break-up, and for µ > µ T all equilibria are unstable.

Slow passage problem.
In this section, we study the slow passage passing through multiple bifurcation points and investigate the delayed onset conditions of a limit cycle, 2-tori and transient behavior, with fixed ν = 0.5 and µ slowly increased with a linear ramp through the various bifurcations described in the previous section.
Figure 2 shows typical evolutions with µ 0 = 0.25 and two different ramp rates = 10 −3 and 10 −5 (several over a wide range have been considered), and Fig. 3 shows typical evolutions with = 10 −4 and different µ 0 ∈ [0.25, 0.55].In all cases, the trajectories initially collapse towards the origin P 1 with (r 1 = 0, r 2 = 0), and then after µ has increased beyond µ H2 , the trajectories rapidly jump to P 2 (which has r 1 = 0 and r 2 = 0).This jump is rapid but smooth; the inflection point in r 1 (µ(t)) at the jump is used to define the first jump point µ = µ 1 .Note that µ 1 > µ H2 , and µ 1 − µ H2 is the delay in observed P 2 oscillations.As noted in [1,13], when = 0 and → 0, the onset of Hopf oscillations is delayed until the control parameter has been increased past the Hopf critical value by an amount equal to how far below the critical value the parameter was at the start of the linear ramp, that is ) Figure 4 illustrates this delay scaling law for our problem with = 10 −5 .
They [1,13] also noted that the delay effect (the jump condition) was independent of the slow ramp rate .Both of these conditions are also evident in our problem, where we have considered several decades of variation in the ramp rate.Figure 5 includes the variation with of the first jump µ 1 to the P 2 Hopf oscillations for µ 0 = 0.42, showing independence for < 10 −3.5 .
To analytically consider the oscillation onset condition µ 1 for linear ramps µ(t) = µ 0 + t, the following general condition [10,11] is very useful: where {Reλ(µ(t))} max denotes the real part of the eigenvalue with largest real part computed from the linearization of the differential equation system with µ fixed at its value at time t, and µ i is the ith jump value of µ.
For the first jump condition, µ 1 , λ 1 (t) = − 11 7 ν + 75 56 µ(t) should be used in (12), and since it is linear in t it is straightforward to find t * satisfying and it is t * = 2(µ H2 − µ 0 )/ .Therefore, the first jump condition satisfying (11) is 3.2.Slow passage through the Neimark-Sacker bifurcation point µ NS .As µ continues to increase past µ NS , the critical value corresponding to the Neimark-Sacker bifurcation when = 0, in the slow passage problem the trajectory continues to track very close to the now unstable limit cycle P 2 before jumping to the 2-torus P 4 , resulting in a secondary delay effect.Again, the jump is rapid but smooth and we use the inflection point at µ = µ 2 to define this second jump.The delay µ 2 −µ NS is also evident in Fig. 3, which indicates that the delay in the jump to the 2-torus P 4 follows a similar scaling law to the first delay (11): (a) µ 0 = 0.25  To derive the onset condition for the second jump from the limit cycle P 2 to the  it is t * = 2(µ NS − µ 0 )/ and Figure 6 numerically shows this secondary delay scaling law, which resulted from considering many starting points µ 0 , both greater and smaller than the Hopf point µ H2 .As with the delay in the Hopf oscillations, the delay in the torus oscillation as → 0 is also independent of .Figure 5 also shows this for µ 0 = 0.42; note that for the second jump µ 2 for the delay in the Neimark-Sacker torus oscillations, needs to be about an order of magnitude smaller than it does for the Hopf delay in order to observe independence.
The two jump conditions (11) and ( 15) in the limit of independence, give the distance between the two jump points: which is twice the distance between the Hopf and Neimark-Sacker bifurcation points.This is independent of the ramp rate , but more important it is also independent of the initial value of µ(t = 0) = µ 0 ; Fig. 3 illustrates this for a few examples.This implies that in a slow passage setting, the Hopf oscillations can be observed for twice the extent of the control parameter µ than would be the case in a static-parameter setting.In an experimental situation, such as in fluid dynamics with a quasi-static parameter variation, Hopf oscillations may be observed for a parameter range that is twice as large as would be expected from a static-parameter stability analysis.

3.3.
Torus break-up and the escape to infinity.As µ(t) continues to be slowly increased beyond the second jump µ 2 , we find a third jump corresponding to the transition from the 2-torus P 4 to infinity.In the = 0 problem, P 4 loses stability at µ = µ T and all the equilibria of ( 1)-( 4) are unstable.In the slow passage problem the trajectory continues to track very close to the unstable P 4 for a long time before eventually jumping away.The value µ = µ 3 is where the slow passage trajectory makes the sudden jump away from P 4 .However, this delay-effect jump behaves very differently to the two jumps associated with the Hopf and Neimark-Sacker bifurcations described earlier.It is not dependent on the initial value µ 0 of µ(t).Also, in contrast to µ 1 and µ 2 which are independent for small enough , this delay scales with the square root of the ramp rate .Figure 7 shows how the delay, µ 3 − µ T , scales with √ ; the many cases shown fall approximately on a line of slope 1/2 in the log-log plot.The ensemble-average scaling law for this jump is The escape from the unstable P 4 can take a very long time in general (not just in the slow passage problem) as the escape route is very narrow in phase space.For µ < µ T , the unstable manifold from P 2 coincides with the stable manifold of P 4 , and the stable manifold of P 3 comes in from infinity.Whereas, for µ > µ T , the unstable manifold from P 2 and the stable manifold of P 4 no longer coincide; the unstable manifold from P 2 now goes off to infinity while the the unstable manifold from P 4 and the stable manifold of P 3 now coincide.Figure 8 illustrates the phase portrait at µ = 2.0 > µ T , showing the stable and unstable manifolds described above that form a very narrow escape tunnel as black curves, one ending at P 3 with (r 1 = 0, r 2 ≈ 3.582) and the other ending at P 2 with (r 1 ≈ 5.766, r 2 = 0).The red (gray) trajectory in the figure is from a slow passage starting very near the origin (r 1 = r 2 = 0.01) with ν 0 = 0.7 and = 10 −4 .
The escape time is (t 2 − t 1 ), where Subtracting ( 20) from (21) gives and averaging over many ensembles, gives an estimate of the escape time: So, using a quasi-static parameter variation (i.e., 0 < << 1), the 2-torus will appear to be stable as the escape time from the neighborhood of the 2-torus is very long, of order 1/ √ .This result is related to a general result on transient dynamics induced by a random perturbation [36].In transient dynamics, it takes a certain time, or probability, for a trajectory to arrive at the entrance of the escape tunnel.In the slow passage problem, after the escape tunnel is created, the time that a spiraling trajectory arrives at the entrance of the tunnel is found to depend on the ramp rate.The slow passage problem through an escape tunnel can thus be considered in terms of random transient dynamics.
For small , the scaling law (19) implies that the third jump µ 3 is very close to µ T .This means that the observed 2-torus P 4 oscillations will be much reduced due to the P 2 Hopf oscillations being observed for twice the parameter extent as indicated by (18). Figure 2 illustrates the reduced observation of the P 4 torus oscillations as is reduced.
4. Conclusions.In this paper, we have considered a slow passage problem in a dynamical system in which there are multiple instabilities amongst a variety of equilibria, including fixed points, limit cycles and 2-tori.In many practical systems, multiple bifurcations and instabilities are encountered as a parameter is varied.When the control parameter is slowly varied in such systems, typical delay and memory effects may be expected.Our work is motivated by the question of whether the delay and memory effects due to slow parameter variations predicted from models that only have a single bifurcation apply to more complex situations with multiple bifurcations that are often encountered in practical problems.By slowly varying the control parameter, we find that when the control parameter passes through the Hopf bifurcation, the delay and memory effects and the onset condition predicted by slow passage problem in the Hopf normal form [13] are confirmed.Further increasing the control parameter beyond the jump to the limit cycle and through the Neimark-Sacker bifurcation, the jump condition depends on the how far from the Neimark-Sacker critical value of the parameter the slow varying parameter is initially, a result that is very similar to the Hopf delay.In both cases, the slow passage characteristics are independent of the ramp rate of the parameter for small enough ramp rate.An interesting result is the slow passage past a Hopf followed by a Neimark-Sacker bifurcation is that the Hopf oscillations are observed over a range of the parameter that is twice that predicted from a static bifurcation analysis, and this result is independent of both the initial parameter value (no memory effect) and the ramp rate.The delay effect for the jump from the unstable 2-torus is also memory-less, not depending on the initial value of the ramped parameter, but it does depend on the ramp rate.This rather unexpected behavior can be understood in terms of the formation of an escape tunnel away from the unstable 2-torus from the stable and unstable manifolds of the associated limit cycles.In the problem investigated, the limit cycle P 3 remains unstable throughout the whole range of the control parameter and superficially seems not to play any role in the slow passage problem as the trajectories are never close to it.However, the slow-passage jump from the 2-torus is critically controlled by P 3 as its stable manifold sets up the escape tunnel and is responsible for the square-root of the ramp rate dependence of the jump condition, in stark contrast to the jump conditions on the slow passage past the Hopf and the Neimark-Sacker bifurcations which do not depend on the ramp rate.
Our results on slow passage through multiple bifurcations may have strong implications for problems where similar bifurcations are found in a very narrow range of parameters.For example, in rotating convection the onset of spatio-temporal chaos of Küppers-Lortz type was shown to occur following a succession of symmetrybreaking Hopf and secondary Hopf bifurcations, akin to the bifurcation sequence in the model problem studied here, over a parameter variation of about 2% [37].Slow drifts in control parameters over the very long times needed near the onset of Küppers-Lortz type of instability in a physical experiment may need to include considerations from the present study.Another hydrodynamics problem in which a sequence of symmetry-breaking Hopf bifurcations culminating in a torus break-up leading to spatio-temporal chaos as a control parameter is varied by about 2% is harmonically forced Taylor-Couette flow [38].In experimental studies of that flow [39], the parameter is very difficult to keep constant to the level of precision required, and again, the slow passage results from this study may need to be considered in interpreting observations.

Figure 1 .
Figure 1.(Color online) Bifurcation diagrams with ν = 0.5; stable branches are solid lines and unstable branches are dashed lines.The critical points where solution branches are created or change their stability are indicated as µ H1 and µ H2 (Hopf bifurcations), µ NS (Neimark-Sacker bifurcation), and µ T (torus break-up).

Figure 2 .
Figure 2. Evolution of r 1 and r 2 for fixed ν = 0.5 as µ is slowly varied according to µ(t) = 0.25 + t, with as indicated.The bifurcation diagram from Fig. 1 is reproduced as faint gray lines to guide the eye.

Figure 3 .
Figure 3. Evolution of r 1 and r 2 for fixed as µ is slowly varied according to µ(t) = µ 0 + 10 −4 t, with µ 0 as indicated.The bifurcation diagram from Fig. 1 is reproduced as faint gray lines to guide the eye.

Figure 8 .
Figure 8. (Color online) Phase portraits, showing the stable manifold of P 3 and the unstable manifold of P 2 in black for ν = 0.5 and µ = 2.0, these form a narrow escape tunnel, along with a slow passage trajectory (red/gray) starting near the origin with ν 0 = 0.7 and = 10 −4 .