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Chaotic systems are ones for which small changes eventually lead to results that can be
dramatically different. The Rössler system is one of the simplest sets of differential equations that
exhibits chaotic dynamics. In addition to their theoretical value in studying chaotic systems, the Rössler
equations are useful in several areas of physical modeling including analyzing chemical kinetics for
reaction networks. Consider the reaction network:
A2+x22x
Az
X+Y2Y
Az+YA
ky
X+2=A,
-Y-2
A4+2 =22
where X, Y, and Z represent the chemical species whose concentrations vary and A., A2, A3, A., and As
are chemical species whose concentrations are held fixed by large chemical reservoirs, serving to keep
the system out of thermodynamic equilibrium. K, and k_, denote the forward and inverse reaction rates.
The system of differential equations that describe the concentrations x, y, and z (for chemical species X,
Y, and Z) are:
dx
dt
dy
= x + ay
dt
dz
=b-C2+ XZ
dt
a) Simulate this system for a=0.380, b = 0.300, and c = 4.280 with initial conditions x(0) = 0.1,
y(0) = 0.2, 7(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 x, y, and z versus time on one figure with three
subplots. Additionally, in separate graphs, plot the phase-space plots: x versus y, x versus 2, and y
versus z. Finally, make a 3-D plot of x vs y vs z 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 (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 x vs t, y vs t, and z vs t. In addition, make plots of the differences:
x(t) – Warsinailt) vs t, y(t) - Yorgina(t) vst, and z(t) – Zorgina(t) vs t.
c) Illustrate the sensitivity of the solution to variations in parameter values by repeating the simulation
of part (a) with c = 4.280001. [Use the original initial conditions from part (a).] Show the sensitivity
with the same set of plots as in part (b).
Why is this system non-linear?
Qualitatively describe the sensitivity to initial conditions and parameter values.
Chaotic systems are ones for which small changes eventually lead to results that can
dramatically different. The Rössler system is one of the simplest sets of differential equations tha
exhibits chaotic dynamics. In addition to their theoretical value in studying chaotic systems, the F
equations are useful in several areas of physical modeling including analyzing chemical kinetics fc
reaction networks. Consider the reaction network:
A. +X=2x
X+Y=2Y
AstY ZA
X+2=A,
ke
Ax+2=22
where X, Y, and represent the chemical species whose concentrations vary and A, A2, A3, Aa, ar
are chemical species whose concentrations are held fixed by large chemical reservoirs, serving to
the system out of thermodynamic equilibrium. k, and k_, denote the forward and inverse reactio
The system of differential equations that describe the concentrations x, y, and z (for chemical spe
Y, and 2) are:
dx
=-y-Z
dt
dy
-= x + ay
dt
dz
-=b-c2 + x2
dt
a) Simulate this system for a=0.380, b = 0.300, and c = 4.280 with initial conditions (0) = 0.1,
y(0) = 0.2, 2(0) = 0.3. Run the simulation for 200 seconds using a fixed-step size algorithm wit
size of 0.001 seconds. Plot the concentrations x, y, and z versus time on one figure with three
subplots. Additionally, in separate graphs, plot the phase-space plots: x versus y, x versus z, a
versus z. Finally, make a 3-D plot of x vs y vs z 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 valu
the initial condition in either direction.) Keep the initial conditions for y(0) and z(0) the same as
part (a). Show the sensitivity by superimposing the plots for the new values you obtain for x(t),
and z(t) with the original plots for x vs t, y vs t, and z vs t. In addition, make plots of the differeni
x(t) – orginai(t) vs t, y(t) – Yorginal(t) vs t, and z(t) – Zorgina(t) vs t.
c) illustrate the sensitivity of the solution to variations in parameter values by repeating the simuli
of part (a) with c = 4.280001 . [Use the original initial conditions from part (a).] Show the sensitiv
with the same set of plots as in part (b).
Why is this system non-linear?
Qualitatively describe the sensitivity to initial conditions and parameter values.