Exponentially stable fixed point

Using the same idea as above, we can also construct prototypical examples of transcritical and pitchfork bifurcations at a stable fixed point. In the x-direction the dynamics are given by the normal forms discussed in Chapter 3, and in the y-direction the motion is exponentially damped. This yields the following examples ... 

Assuming Sxi = (5xo exp[(t - 1)A] for large i, we may compute A via A = limi cc lnX5,=o 4 (1 - 2x ). Figure .12 shows A = A(r). Negative values mean that the iteration approaches a stable fix point or limit cycle. Positive values mean that a small perturbation grows exponentially. We notice that the bifurcations are associated with A = 0. A is called Lyapunov-exponent. 

Hence x(f) is a combination of terms involving e cosdX and e sin6jf. Such terms represent exponentially decaying oscillations if a = Re(A) < 0 and growing oscillations if a > 0. The corresponding fixed points are stable and unstable spirals, respectively. Figure 5.2.4b shows the stable case. 

The previous analysis is confirmed when one measures the quasiclassical probability Pn(t) for remaining in the initial state (n, 0) for an ensemble of two hundred trajectories defined with initial conditions J = n + 1/2, = random J = 1/2, = random. The decay is close -fo exponential when the central fixed point is unstable and on the contrary it is distinctively non exponential with prominent oscillations ("beats") when the fixed point is stable. Thus, the sensitivity of short time relaxation to potential energy coupling derives from the drastic effect of a change from instability to stability on the stretch-bend energy flow in quasiperiodic trajectories. In this way, we have shown that the short time overtone decay dynamics of the two mode model exhibit the same sensitivity to potential energy coupling as does the trajectory calculations for the full planar benzene Hamiltonian of Hase and coworkers(10). 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

In the case of trirhythmicity, for example, when the initial values of and y are fixed and only the initial concentration of the substrate, a, is changed, we observe (table 4.3) that the system evolves, alternatively, towards the cycle LCl or LC2. The intervals of a values corresponding to these successive choices diminish exponentially as a increases, until an accumulation point is reached, beyond which the system evolves towards the third cycle, LC3. The origin of such behaviour, also observed in a different context by Takesue Kaneko (1984), can be elucidated by means of one-dimensional maps (Decroly Goldbeter, 1985 Decroly, 1987a). A prediction of this analysis, verified by numerical simulations, is that from certain initial conditions the system can evolve to one of the stable cycles after passing, successively, in a transient manner, by each of the two unstable cycles. 

---