The dynamic model for a certain chemical reaction is given by
xl = s (x2 — x2x1 + xi — qxn ±2 = S-1 (-x2 X2X1 Ex3) ±3 = w(x1 - x3))
here s = 77.27, w = 0.1610, q = 8.375 x 10-6, and E = 1. (A) Verify that the system has an equilibrium point when xi = x3 = 488.68 and
x2 = 0.99796. (B) Linearize the system about the four points xA -= (1 10,000 1,000 )T xB = ( 31,623 10 100.000 )T xc = ( 1 1 1 )T and the equilibrium point. Calculate the eigenvalues and stiffness ratios of the linearizations. (C) Simulate the system for at least 400 seconds for the initial conditions xA, xB, xc, xss+ and xss_ where xss+ is an initial condition near the equilibrium point for which x1 increases and where xss_ is an initial condition near the equilibrium point for which x1 decreases. Provide the following plots: (1) A plot with three subplots showing each logm(xj) plotted versus time for the xc initial condition. (2) A plot with logio(x2) on the horizontal and logio(xi) on the vertical for all five initial condition responses, each response in a different color. (3) A plot with loglo(x2) on the horizontal and logio(x3) on the vertical for all five initial condition responses, each response in a different color.
Note that all the simulations should be stable.
The Van Der Pol system is given by the state-space representation
i
= x2
2 = E(1 - X0X2 - Xi ± U,
where u is the forcing function. Let E be fixed to e = 5, and obtain the phase-plane plots for the forced Van Der Pol system with the input u given by
u(t) = a cos (wt)
for the pairs of magnitudes and frequencies given in Table 1. You are encouraged to explore other magnitudes close to the ones given in the table. Use at least two different integration methods to obtain your results (over all the sets of parameters, I am not expecting you to simulate each set of parameters more than once), and discuss how and why you chose the integration method and time-steps that you used.
Table 1: Parameter Values for Force Van Der Pol Equation. a w
15 25 7 7 50 7 55 7 1 3 5 3
A dynamic model of love was formulated by Rinaldi et al5 and applied to the Disney film Beauty and the Beasts as follows. The state variables are the feelings at the time t of the individuals for their partners. The dynamic model is
(t) = —a fr (t) + (x2) + A2(t) x2(t) = —a2x2(t) + R2 (Z1) + Al (t)
where
a1 = 0.1, al = 0.3, , e'2 — Ri (x2) —
and The next equation was modified.
(1)
2e5' — 2e-x' R2(xi) el + 2e-=, • The a1 and a2 parameters have to do with placing higher significance on more recent events. The R1(x2) and R2(xi) are descriptions of how one person reacts to the feelings of the other person. The final pair of parameters Al(t) and A2(t) represent the appeal of one person to the other. In Beauty and the Beast appeal of Belle is Ai (t) high and a constant, let Ai(t) = 1.2. The appeal of the Beast will change with time, and make the love story possible. Note that xi (t) represents the feelings that Belle has for the Beast and that x2(t) represents the feelings that the Beast has for Belle. Assume that the two start with indifferent feelings toward each other, that is x1(0) = x2(0) = 0. (A) Initially, the appeal of the Beast is very negative because of his, well, bestial appearance and behavior. For the first simulation, keep the appeal of the Beast constant at A2 = —1.9, and simulate the resulting system until a steady state is reached. (B) In the story, the Beast becomes less unappealing, i.e. A2(t) increases, through his actions of saving Belle from the wolves, allowing Belle into his library, etc. We will represent this scenario by setting
A2(t) = 0.02t — 1.9, 0 < t S 35 —1, 35 < t Though the Beast becomes more appealing, in this scenario Belle is still frightened by the Beast's bestial appearance. Simulate this scenario until some steady state has been reached. (C) Repeat part (B), except use
A2(t) = 0.02t — 1.9, 0 < t < 80 - 0.61 80 < t •
In this scenario, Belle is more able to look past the Beast's appearance, but there is still some revulsion to it.
(D) Briefly discuss the three scenarios, and suggest which of the scenarios fits the story best. Spoiler Alert: In the end, Belle and the Beast fall in love and live happily ever after.
sltinaldi at all, "Love and appeal in standard couples." International Journal of Bifurcation and Chaos, 2000. 6Rinaldiet al, "Small discoveries can have great consequences in love affairs: the case of Beauty and the Beast." International Journal of Bifurcation and Chaos, 2013.
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