Mathematics 3DC3 ASSIGNMENT 2
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Mathematics 3DC3
ASSIGNMENT 2
Due Wednesday, February 15, 2023 11:50pm on Crowdmark. Late assignments will not be accepted without prior agreement.
1. Consider the map F入 (x) = λx - x3 .
(a) Find all of the fixed points and all of the bifucations of these fixed points analytically and deter- mine if the bifurcations are saddle-node, pitchfork, transcritical, or period doubling bifurcations. Determine where the fixed points are attractive and where they are repulsive. Draw by hand, phase portraits for values of the bifurcation parameter at and on each side of the bifurcations of the fixed points.
(b) Find the curves of any 2-cycles analytically as functions of the parameter. Also plot the function and the second iterate of the function for values of the parameter between the critical values of the bifurcations to determine that you have found all of the prime 2 cycles. Determine where the 2-cycles are attractive and where they are repulsive analytically.
(c) i. Draw by hand a bifurcation diagram for λ e [-2.1, 2.2] indicating curves of stable fixed points or periodic points with solid curves and unstable curves with dashed curves.
ii. Using computer software, provide an orbit diagram for λ e [-2, 4]. Interpret what is happening at λ = -2,and at the first 3 bifucations of λ > 0. Does your orbit diagram depend on your choice of initial condition for x?
(Hint: You might try to include as one of the option commands in XPPAUT:
@ bounds = 1000000 to “try” to avoid an “out of bounds error” message. You can also do this from within XPPAUT using “NUMERICS-→BOUNDS” and typing 1000000 in the blank space at the top.)
iii. Using computer software, provide a bifurcation diagram for λ e [-2, 3.5]. (Make sure none of your conclusions contradict each other (i.e., the analysis and the numerics agree)!)
2. Consider the map fc (x) = x + c sin(x).
(a) Find all of the fixed points and all of the bifucations of these fixed points analytically and deter- mine if the bifurcations are saddle-node, pitchfork, transcritical, or period doubling bifurcations.
Determine where the fixed points are attractive and where they are repulsive. (b) Consider the bifurcation diagram shown in Figure 1.
|
|
2
0
-2
-4
-6
-3 -2 - 1 0 1 2 3 4
c
Figure 1: Bifurcation diagram for fc (x) = x + c sin(x).
Using a combination of analysis and numerics, explain with justification the bifurcations shown. Hint: Which ones are pitchfork bifurcations of the map, period doublings of the map, pitchork bifurcations of a higher iterate of the map or period doublings of a higher iterate of the map. In particular, include an orbit diagram starting from different initial conditions for c e [-3.5, 4] and x e [-7, 7]. Freeze with different colours to try to include all the attractive invariant sets that are shown in the bifurcation diagram.
Hints:
❼ You will need to start from different seeds.
❼ You need to choose a negative number from -1 to -10 when you freeze.
❼ Does your orbit diagram support your explanations in the previous part. If not, figure out what ou did wrong and revise.
2023-03-24