MATLAB and Simulink Chaotic Systems

cct1987
timer Asked: Mar 30th, 2015

Question Description

Document1.pdf This problem is supposed to be solved using MATLAB. Please provide m files and simulink models and screenshots of system and resulting plots.


Unformatted Attachment Preview

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.
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

This question has not been answered.

Create a free account to get help with this and any other question!

Related Tags

Brown University





1271 Tutors

California Institute of Technology




2131 Tutors

Carnegie Mellon University




982 Tutors

Columbia University





1256 Tutors

Dartmouth University





2113 Tutors

Emory University





2279 Tutors

Harvard University





599 Tutors

Massachusetts Institute of Technology



2319 Tutors

New York University





1645 Tutors

Notre Dam University





1911 Tutors

Oklahoma University





2122 Tutors

Pennsylvania State University





932 Tutors

Princeton University





1211 Tutors

Stanford University





983 Tutors

University of California





1282 Tutors

Oxford University





123 Tutors

Yale University





2325 Tutors