Prove that the invariant measure of the chaos generated by the map f (x) is μ(x) ≡ 1,and convincingly prove that the Lyapunov exponent is λ = −a ln(a) − (1 − a) ln(1 − a)
Math/Physic/Economic/Statistic Problems
1. Consider the system
(xn+1 yn+1)= L(f (xn)f (yn), with L =[1 − ǫ ǫǫ 1 − ǫ]
. (1) and f (x) ={x/a, 0 ≤ x ≤ a(1 − x)/(1 − a), a < x ≤ 1. (2)
Prove that the invariant measure of the chaos generated by the map f (x) is μ(x) ≡ 1,and convincingly prove that the Lyapunov exponent is λ = −a ln(a) − (1 − a) ln(1 − a)
2. Now iterate this system on the computer for a = 0.7 and ǫ = 0.2, which is below the critical coupling strength ǫc. Show a picture of the outbreaks. Sometimes, for some of you, the system can accidentally get stuck into the synchronized state, which it never leaves. Why ? Think about numerical precision. 1 2
3. Avoid this being trapped inside the computer by taking the parameters a slightly different for the two coupled maps. Compute the histogram of outbreaks P (z), z =ln |(xn − yn)/2|, and verify its exponential behavior, P (z) ∝ exp(κz). As explained in the lecture notes, κ should be proportional to the transverse Lyapunov exponent λ⊥
(for which an analytical expression exists: Eq.8.4 of the lecture notes). You will report on this proportionality by running the systems for various coupling strengths, both below and above criticality. The proportionality factor involves the second (curvature) moment of the distribution of the finite-time Lyapunov exponents λ of the individual
system. You can either compute the analytic form (Eq.8.14 of the lecture notes), or find it from simulating the (single) map and looking at the distribution of λ. You must try very hard to verify this relation.
Maps are quick, but it is interesting to see this theory work for differential equations. It also provides confidence in handling continuous-time systems through integration of coupled ordinary differential equations. You will find that the phenomenology is exactly the same, so, if you understood the previous questions, you will know exactly what to do in the ODE system.
d2θmdt2 + γ dθmdt + sin θm = Γ0 cos(Ωt) + c(sin θm − sin θs)/2 (3)
d2θsdt2 + γ dθsdt + sin θs = Γ0 cos(Ωt) + c(sin θs − sin θm)/2 (4)
with damping γ = 0.2, Γ0 = 1.2 and Ω = 0.5. It is the synchronization of symmetrically coupled driven pendula. It so happens (which you may verify) that the critical coupling strength ccr = 0.7948 . . ..
4. Compute the histogram of the difference between slave and master, z(t) = ln({(θs −θm)2 + ( ̇θs − ̇θm)2}1/2) slightly below criticality (ccr = 0.7948). Avoid the “getting stuck” pitfall. Show a picture of the histogram. Also realize that θ and θ + 2π are exactly the same state.
5. Now we must verify from the value of the transverse Lyapunov exponent λ⊥ that the number above is indeed the critical coupling strength. Compute λ⊥ and show that it changes sign if we move through ccr. See the section “how to compute Lyapunov exponents”.
First prove that the transverse dynamics δ⊥ = θm − θs satisfies the linear equation
̈δ⊥ + γ ̇δ⊥ + (1 − c)δ⊥ cos θ(t) = 0,
1This has resulted in great confusion in the literature, see the remarkable paper Reconsideration of intermittent synchronization in coupled chaotic pendula, by S. Rim et al., Phys. Rev. E 64, 060101(R) (2001). See also the references at the end of Chapter VIII of the lecture notes. The articles are on Brightspace.
2MATLAB has an option to compute with more than 16 digits, but this has little to do with physics with driving force cos θ(t) from the synchronized state. Add this ODE to your system of ODE’s (see the last page of this assignment) and use it to compute the Lyapunov exponent λ⊥.
In order to evade the locking pitfall, you should add noise. For example, uniformly distributed random numbers on the interval [−5 × 10−11, 5 × 10−11] to θm and θs after each 10 integration steps (0.1 s). Of course this implies that z can never become smaller than ≈ 10−10. In this sense, synchronization is the ability of the coupled systems to overcome tiny noise that tries to push them apart while it is amplified by their positive Lyapunov exponent.
6. Finally, establish the relation between the “shape” κ of the histogram of outbreaks and λ⊥.
Do the same as in (3), i.e. relate the proportionality factor between κ and λ⊥ to the width of to the Gaussian distribution of long-time Lyapunov exponents ΛT . This is an “open problem”, you can go as far as you like. Ultimately you would like to see the same relation between the statistics of the outbreaks, the value of the transverse Lyapunov exponents, and the statistics of the instantaneous “Lyapunov exponents” of the synchronized system as in the coupled map case. Lyapunov exponents.
