Nonlinear dynamical systems.
Bakground: The goal of this excercise is to investigate theoretically and experimentally how the dynamical properties of a map or di erential equation depend on the system parameters.
1. For a and b real parameters, consider the HØnon map:
( 2 xn+1 = 1+ yn − axn
yn+1 = bxn
Question 1 Where in (b,a)-plane do xed points exist? Compute the region of stability in (b,a)-plane where at least one xed point is stable. See a detailed hint at the end of this list.
Question 2 Let b = 0.3 throughout and investigate (numerically) what happens when a varies from 0 to 3.
Question 3 For b = 0.3, build up the so-called bifurcation diagram by plotting the x variable of the stationary solution against the parameter a. Plot just points, not lines joining the points.
2. For the di erential equation:
Question 4 (a) Find the xed point. Compute its stability region in (b,a)plane. (b) For which (b,a)-values the system has a periodic orbit? Where is it stable? (Hint: Use polar coordinates).
Question 5 (a) Plot a few orbits of the system for a = 1 to get an idea of the overall dynamics. (b) Same for a = −1.
Question 6 Let a = b = 1 and nd a trapping region. Note that in this case the origin is unstable, so you need to nd a large region of phase space such that the ow points inwards on the surface of the region.
Hint to stability part of Question 1
1. First compute the x-coordinate of the xed points x±. Note that for a > 0 both xed points have di erent signs on their x-coordinate.
2. Compute the jacobian as a function of x±, note Trace and Determinant.
3. Appendix A.4.6.2(b) Compute the necessary condition for sability interms of b.
4. Compute the eigenvalues of xed points in terms of x:
λi(x) = −ax + ···.
5. To get stability, in addition to the necessary condition on b, it is required that the eigenvalue of largest modulus for at least one xed point satis es |λMAX(x)| < 1. The border of the stability region is |λMAX(x)| = 1.
6. You get two conditions |λMAX(x+)| = 1, |λMAX(x−)| = 1. One of them can be discarded, while the other gives the border of a region in the (a,b)-plane. Check if the relevant part is the interior region or the exterior region (pick one easy point (a0,b0) and check xed points, stability, eigenvalues, etc.).
