# Sysen 533 exam 1 | Physics homework help

 SYSEN 533 – Penn Exam 1 Page 2 of 4

2. (25 pts) Chaotic systems areö ones for which small changes eventually lead to results that can be dramatically different. The R ssler system is one of the simplest sets of differential equations thatöexhibits chaotic dynamics. In addition to their theoretical value in studying chaotic systems, the R ssler equations are useful in several areas of physical modeling including analyzing chemical kinetics for reaction networks. Consider the reaction network:

k1

A2X

k1

k2

+2Y

k2

k3

A5+Y A2

k3

k4

+Z A3

k4

k5

A+2Z

k5

where XY, and represent the chemical species whose concentrations vary and A1A2A3A4, and Aare chemical species whose concentrations are held fixed by large chemical reservoirs, serving to keep the system out of thermodynamic equilibrium. kand kdenote the forward and inverse reaction rates.

The system of differential equations that describe the concentrations xy, and (for chemical speciesXY, and Z) are:

 dx =−y −z dt dy = x +ay dt dz =b −cz +xz dt

a)Simulate this system for a=0.380, = 0.300, and = 4.280 with initial conditions x(0) = 0.1,

y(0) = 0.2, z(0) = 0.3. Run the simulation for 200 seconds using a fixed-step size algorithm with a step size of 0.001 seconds. Plot the concentrations xy, and versus time on one figure with three subplots. Additionally, in separate graphs, plot the phase-space plots: versus yversus z, and y versus z. Finally, make a 3-D plot of vs vs using the Matlab graphics command “plot3”

b)Illustrate the sensitivity of the solution to variations in the initial conditions by repeating the simulation of part (a) with x(0) = 0.0999 and then with x(0) = 0.1001 . (A 0.1% change in the value of the initial condition in either direction.) Keep the initial conditions for y(0) and z(0) the same as in part (a). Show the sensitivity by superimposing the plots for the new values you obtain for x(t),y(t), and z(t) with the original plots for vs tvs t, and vs t. In addition, make plots of the differences: x(t) – xorginal(t) vs ty(t) – yorginal(t) vs , and z(t) – zorginal(t) vs t.

c)Illustrate the sensitivity of the solution to variations in parameter values by repeating the simulation of part (a) with =4.280001 . [Use the original initial conditions from part (a).] Show the sensitivity with the same set of plots as in part (b).

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