Difference between revisions of "Dynamic Behavior"
m (→Chapter Summary) 

(21 intermediate revisions by the same user not shown)  
Line 5:  Line 5:  
{{chaptertable left}}  {{chaptertable left}}  
== Textbook Contents ==  == Textbook Contents ==  
−  {{am05pdfam08dynamics  +  {{am05pdfam08dynamics28Sep12Dynamic Behavior}} 
* 1. Solving Differential Equations  * 1. Solving Differential Equations  
* 2. Qualitative Analysis  * 2. Qualitative Analysis  
Line 29:  Line 29:  
This chapter introduces the basic concepts and tools of dynamical systems.  This chapter introduces the basic concepts and tools of dynamical systems.  
<ol>  <ol>  
−  <li> <p> We say that <  +  <li> <p> We say that <amsmath>x(t)</amsmath> is a solution of a differential equation on the time interval <amsmath>t_0</amsmath> to <amsmath>t_f</amsmath> with initial value <amsmath>x_0</amsmath> if it satisfies 
<center><amsmath>  <center><amsmath>  
x(t_0) = x_0 \quad\text{and}\quad \dot x(t) = F(x(t)) \quad\text{for all}\quad t_0 \leq t \leq t_f.  x(t_0) = x_0 \quad\text{and}\quad \dot x(t) = F(x(t)) \quad\text{for all}\quad t_0 \leq t \leq t_f.  
</amsmath></center>  </amsmath></center>  
−  We will usually assume <  +  We will usually assume <amsmath>t_0 = 0</amsmath>. For most differential equations we will encounter, there is a unique solution for a given initial condition. Numerical tools such as MATLAB and Mathematica can be used to obtain numerical solutions for <amsmath>x(t)</amsmath> given the function <amsmath>F(x)</amsmath>.</p></li> 
−  <li><p> An ''equilibrium point'' for a dynamical system represents a point <  +  <li><p> An ''equilibrium point'' for a dynamical system represents a point <amsmath>x_e</amsmath> such that if <amsmath>x(0) = x_e</amsmath> then <amsmath>x(t) = x_e</amsmath> for all <amsmath>t</amsmath>. Equilibrium points represent stationary conditions for the dynamics of a system. A ''limit cycle'' for a dynamical system is a solution <amsmath>x(t)</amsmath> which is periodic with some period <amsmath>T</amsmath>, so that <amsmath>x(t + T) = x(t)</amsmath> for all <amsmath>t</amsmath>.</p></li> 
<li><p><span id=stability>An equilibrium point is (locally) ''stable'' if initial conditions that start near an equilibrium point stay near that equilibrium point. A equilibrium point is (locally) ''asymptotically stable'' if it is stable and, in addition, the state of the system converges to the equilibrium point as time increases. An equilibrium point is ''unstable'' if it is not stable. Similar definitions can be used to define the stability of a limit cycle.</span></p></li>  <li><p><span id=stability>An equilibrium point is (locally) ''stable'' if initial conditions that start near an equilibrium point stay near that equilibrium point. A equilibrium point is (locally) ''asymptotically stable'' if it is stable and, in addition, the state of the system converges to the equilibrium point as time increases. An equilibrium point is ''unstable'' if it is not stable. Similar definitions can be used to define the stability of a limit cycle.</span></p></li>  
−  <li><p> Phase portraits provide a convenient way to understand the behavior of 2dimensional dynamical systems. A phase portrait is a graphical representation of the dynamics obtained by plotting the state <  +  <li><p> Phase portraits provide a convenient way to understand the behavior of 2dimensional dynamical systems. A phase portrait is a graphical representation of the dynamics obtained by plotting the state <amsmath>x(t) = (x_1(t), x_2(t))</amsmath> in the plane. This portrait is often augmented by plotting an arrow in the plane corresponding to <amsmath>F(x)</amsmath>, which shows the rate of change of the state. The following diagrams illustrate the basic features of a dynamical systems: 
<table border=0>  <table border=0>  
<tr>  <tr>  
Line 46:  Line 46:  
<td align=center width=33%> [[Image:stablepp.png180px]]</td>  <td align=center width=33%> [[Image:stablepp.png180px]]</td>  
</tr><tr>  </tr><tr>  
−  <td align=center>An asymptotically stable equilibrium point at <  +  <td align=center>An asymptotically stable equilibrium point at <amsmath>x = (0, 0)</amsmath>.</td> 
−  <td align=center>A limit cycle of radius one, with an unstable equilibrium point at <  +  <td align=center>A limit cycle of radius one, with an unstable equilibrium point at <amsmath>x = (0,0)</amsmath>.</td> 
−  <td align=center>A stable equlibirum point at <  +  <td align=center>A stable equlibirum point at <amsmath>x = (0,0)</amsmath> (nearby initial conditions stay nearby).</td> 
</tr></table>  </tr></table>  
</p></li>  </p></li>  
Line 56:  Line 56:  
\frac{dx}{dt} = A x  \frac{dx}{dt} = A x  
</amsmath></center>  </amsmath></center>  
−  is asymptotically stable if and only if all eigenvalues of <  +  is asymptotically stable if and only if all eigenvalues of <amsmath>A</amsmath> all have strictly negative real part and is unstable if any eigenvalue of <amsmath>A</amsmath> has strictly positive real part. A nonlinear system can be approximated by a linear system around an equilibrium point by using the relationship 
<center><amsmath>  <center><amsmath>  
\dot x = F(x_e) + \left.\frac{\partial F}{\partial x}\right_{x_e} (x  x_e) +  \dot x = F(x_e) + \left.\frac{\partial F}{\partial x}\right_{x_e} (x  x_e) +  
\text{higher order terms in $(xx_e)$}.  \text{higher order terms in $(xx_e)$}.  
</amsmath></center>  </amsmath></center>  
−  Since <  +  Since <amsmath>F(x_e) = 0</amsmath>, we can approximate the system by choosing a new 
−  state variable <  +  state variable <amsmath>z = x  x_e</amsmath> and writing the dynamics as <amsmath>\dot z = A z</amsmath>. The stability of the nonlinear system can be determined in a local neighborhood of the equilibrium point through its linearization. 
</p></li>  </p></li>  
−  <li><p>A ''Lyapunov function'' is an energylike function <amsmath>V:R^n \to R</amsmath> that can be used to reason about the stability of an equilibrium point. We define the derivative of <  +  <li><p>A ''Lyapunov function'' is an energylike function <amsmath>V:R^n \to R</amsmath> that can be used to reason about the stability of an equilibrium point. We define the derivative of <amsmath>V</amsmath> along the trajectory of the system as 
<center><amsmath>  <center><amsmath>  
\dot V(x) = \frac{\partial V}{\partial x} \dot x = \frac{\partial V}{\partial x} F(x)  \dot V(x) = \frac{\partial V}{\partial x} \dot x = \frac{\partial V}{\partial x} F(x)  
</amsmath></center>  </amsmath></center>  
−  Assuming <  +  Assuming <amsmath>x_e = 0</amsmath> and <amsmath>V(0) = 0</amsmath>, the following conditions hold: 
<center>  <center>  
{ border=1  { border=1  
Line 76:  Line 76:  
    
 align=center  <amsmath> V(x) > 0, x \neq 0</amsmath>   align=center  <amsmath> V(x) > 0, x \neq 0</amsmath>  
−   align=center  <amsmath>\dot V(x) \leq 0</amsmath> for all <  +   align=center  <amsmath>\dot V(x) \leq 0</amsmath> for all <amsmath>x</amsmath> 
−   align=left  <  +   align=left  <amsmath>x_e</amsmath> stable 
    
 align=center  <amsmath>V(x) > 0, x \neq 0 </amsmath>   align=center  <amsmath>V(x) > 0, x \neq 0 </amsmath>  
 align=center  <amsmath>\dot V(x) < 0, x \neq 0</amsmath>   align=center  <amsmath>\dot V(x) < 0, x \neq 0</amsmath>  
−   align=left  <  +   align=left  <amsmath>x_e</amsmath> asymptotically stable 
}  }  
</center>  </center>  
Line 87:  Line 87:  
</p></li>  </p></li>  
−  <li><p>The ''KrasovskiiLaSalle Principle'' allows one to reason about asymptotic stability even if the time derivative of <  +  <li><p>The ''KrasovskiiLaSalle Principle'' allows one to reason about asymptotic stability even if the time derivative of <amsmath>V</amsmath> is only negative semidefinite (<amsmath>\leq 0</amsmath> rather than <amsmath>< 0</amsmath>). Let <amsmath>V:R^n \to R</amsmath> be a ''positive definite function'', <amsmath>V(x) > 0</amsmath> for all <amsmath> x \neq 0</amsmath> and <amsmath>V(0) = 0</amsmath>, such that 
<center>  <center>  
<amsmath>\dot V(x) \leq 0</amsmath> on the compact set <amsmath>\Omega_c = \{x \in R^n:V(x) \leq c\}</amsmath>.  <amsmath>\dot V(x) \leq 0</amsmath> on the compact set <amsmath>\Omega_c = \{x \in R^n:V(x) \leq c\}</amsmath>.  
Line 93:  Line 93:  
Then as <amsmath>t \to \infty</amsmath>, the trajectory of the system will converge to the largest invariant set inside  Then as <amsmath>t \to \infty</amsmath>, the trajectory of the system will converge to the largest invariant set inside  
<center><amsmath>S = \{x \in \Omega_c:\dot V(x) = 0\}</amsmath>.</center>  <center><amsmath>S = \{x \in \Omega_c:\dot V(x) = 0\}</amsmath>.</center>  
−  In particular, if <  +  In particular, if <amsmath>S</amsmath> contains no invariant sets other than <amsmath>x = 0</amsmath>, then 0 is asymptotically stable. 
</p></li>  </p></li>  
Line 99:  Line 99:  
</ol>  </ol>  
−  == Exercises ==  +  {{chaptertable begin}} 
+  {{chaptertable left}}  
+  == Additional Exercises ==  
+  The following exercises cover some of the topics introduced in this chapter. <! Exercises marked with a * appear in the printed text. >  
<ncl>Dynamic Behavior Exercises</ncl>  <ncl>Dynamic Behavior Exercises</ncl>  
== Frequently Asked Questions ==  == Frequently Asked Questions ==  
<ncl>Dynamic Behavior FAQ</ncl>  <ncl>Dynamic Behavior FAQ</ncl>  
== Errata ==  == Errata ==  
−  <ncl>Dynamic Behavior errata</ncl>  +  <ncl>Dynamic Behavior errata v2.11b</ncl> 
+  * [[:Category:Dynamic Behavior errataFull list of errata starting from first printing]]  
* {{submitbug}}  * {{submitbug}}  
+  <! Additional small typos: >  
+  
+  {{chaptertable right}}  
+  == MATLAB code ==  
+  The following MATLAB scripts are available for producing figures that appear in this chapter.  
+  * Figure 4.1: {{matlabfiledynamicsdoscivp.m}}, {{matlabfile.oscillator.m}}  
+  * Figure 4.3: {{matlabfiledynamicsdoscpp.m}}, {{matlabfile.oscillator.m}}  
+  * Figure 4.4: {{matlabfiledynamicsinvpend_phaseplot.m}}, {{matlabfiledynamicsinvpend.m}}  
+  * Figure 4.5: {{matlabfiledynamicslimitcycle.m}}, {{matlabfiledynamicsosc.m}}  
+  * Figure 4.7, 4.8, 4.9: {{matlabfiledynamicsstability.m}}, {{matlabfile.oscillator.m}}, {{matlabfiledynamicssaddle.m}}  
+  * Figure 4.10: {{matlabfiledynamicscongctrl_dynamics.m}}, {{matlabfile.congctrl.m}}  
+  * Figure 4.11: {{matlabfiledynamicslinvsnl.m}}, {{matlabfiledynamicsinvpend.m}}  
+  * Figure 4.14, 4.15: {{matlabfiledynamicsgenswitch_plot.m}}, {{matlabfiledynamicsgenswitch.m}}  
+  * Figure 4.16: {{matlabfiledynamicsinvpend_balanced.m}}, {{matlabfiledynamicsinvpend_bal.m}}  
+  * Figure 4.17: {{matlabfiledynamicspredprey_bif.m}}, {{matlabfiledynamicspredprey.dat}} (generated by {{matlabfile.predprey.mma}}, {{matlabfile.Jac.m}})  
+  See the [[softwaresoftware page]] for more information on how to run these scripts.  
+  
== Additional Information ==  == Additional Information ==  
* [http://www.engin.umich.edu/group/ctm Control tutorials for MATLAB] (U. Michigan)  * [http://www.engin.umich.edu/group/ctm Control tutorials for MATLAB] (U. Michigan)  
+  {{chaptertable end}} 
Latest revision as of 21:40, 23 November 2012
Prev: Examples  Chapter 4  Dynamic Behavior  Next: Linear Systems 
In this chapter we give a broad discussion of the behavior of dynamical systems, focused on systems modeled by nonlinear differential equations. This allows us to discuss equilibrium points, stability, limit cycles and other key concepts of dynamical systems. We also introduce some methods for analyzing global behavior of solutions.
Textbook ContentsDynamic Behavior (pdf, 28Sep12)

Lecture MaterialsSupplemental Information

Chapter Summary
This chapter introduces the basic concepts and tools of dynamical systems.

We say that is a solution of a differential equation on the time interval to with initial value if it satisfies
We will usually assume . For most differential equations we will encounter, there is a unique solution for a given initial condition. Numerical tools such as MATLAB and Mathematica can be used to obtain numerical solutions for given the function . An equilibrium point for a dynamical system represents a point such that if then for all . Equilibrium points represent stationary conditions for the dynamics of a system. A limit cycle for a dynamical system is a solution which is periodic with some period , so that for all .
An equilibrium point is (locally) stable if initial conditions that start near an equilibrium point stay near that equilibrium point. A equilibrium point is (locally) asymptotically stable if it is stable and, in addition, the state of the system converges to the equilibrium point as time increases. An equilibrium point is unstable if it is not stable. Similar definitions can be used to define the stability of a limit cycle.
Phase portraits provide a convenient way to understand the behavior of 2dimensional dynamical systems. A phase portrait is a graphical representation of the dynamics obtained by plotting the state in the plane. This portrait is often augmented by plotting an arrow in the plane corresponding to , which shows the rate of change of the state. The following diagrams illustrate the basic features of a dynamical systems:
An asymptotically stable equilibrium point at . A limit cycle of radius one, with an unstable equilibrium point at . A stable equlibirum point at (nearby initial conditions stay nearby). A linear system
is asymptotically stable if and only if all eigenvalues of all have strictly negative real part and is unstable if any eigenvalue of has strictly positive real part. A nonlinear system can be approximated by a linear system around an equilibrium point by using the relationship
Since , we can approximate the system by choosing a new state variable and writing the dynamics as . The stability of the nonlinear system can be determined in a local neighborhood of the equilibrium point through its linearization.
A Lyapunov function is an energylike function that can be used to reason about the stability of an equilibrium point. We define the derivative of along the trajectory of the system as
Assuming and , the following conditions hold:
Condition on Condition on Stability for all stable asymptotically stable Stability of limit cycles can also be studied using Lyapunov functions.
The KrasovskiiLaSalle Principle allows one to reason about asymptotic stability even if the time derivative of is only negative semidefinite ( rather than ). Let be a positive definite function, for all and , such that
Then as , the trajectory of the system will converge to the largest invariant set inside
. In particular, if contains no invariant sets other than , then 0 is asymptotically stable.
The global behavior of a nonlinear system refers to dynamics of the system far away from equilibrium points. The region of attraction of an asymptotically stable equilirium point refers to the set of all initial conditions that converge to that equilibrium point. An equilibrium point is said to be globally asymptotically stable if all initial conditions converge to that equilibrium point. Global stability can be checked by finding a Lyapunov function that is globally positive definition with time derivative globally negative definite.