Stability of Differential Equations

Most real life problems are modeled by differential equations. Stability analysis plays an important role while analyzing such models. In this project, we demonstrate stability of a few such problems in an introductory manner. We begin by defining different types of stability. Some methods for determining stability of various systems has been studied here. We start our study by categorizing stability of differential equations by the roots of their characteristic equations. Then we discuss the complexities of such analysis. Then we review Lyapunov’s stability concepts which gives a better way to analyze stability of a system of differential equations. We demonstrate the techniques with some examples.

1. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability Analysis of Dierential Equations
Supervisor
Dr. Samir Kumar Bhowmik
Candidates
Abdullah Md. Saifee
S. M. Mustaquim
April 4, 2018
1 / 52

2. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
What is Stability?
By the termstability, we usually meanresistance to
change.
Stable systemmeans
small changes small changes
in=)in
ODE conditions future behavior
2 / 52

3. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
What is Stability?
By the termstability, we usually meanresistance to
change.
Stable systemmeans
small changes small changes
in=)in
ODE conditions future behavior
2 / 52

4. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
What is Stability?
By the termstability, we usually meanresistance to
change.
Stable systemmeans
small changes small changes
in=)in
ODE conditions future behavior
2 / 52

5. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Why study Stability?
Stability is most directly linked with the modeling
processes.
An accurate description of stability of a system can give
insight about its solution states.
Helps understand the features of the system.
3 / 52

6. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Why study Stability?
Stability is most directly linked with the modeling
processes.
An accurate description of stability of a system can give
insight about its solution states.
Helps understand the features of the system.
3 / 52

7. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Why study Stability?
Stability is most directly linked with the modeling
processes.
An accurate description of stability of a system can give
insight about its solution states.
Helps understand the features of the system.
3 / 52

8. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Why study Stability?
Stability is most directly linked with the modeling
processes.
An accurate description of stability of a system can give
insight about its solution states.
Helps understand the features of the system.
3 / 52

9. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Practical Implications
In Physics for example
of an object in a system of objects.
In Economics for example
strategies.
In Biology for example
resources.
and so on.
4 / 52

10. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Practical Implications
In Physics for example
of an object in a system of objects.
In Economics for example
strategies.
In Biology for example
resources.
and so on.
4 / 52

11. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Practical Implications
In Physics for example
of an object in a system of objects.
In Economics for example
strategies.
In Biology for example
resources.
and so on.
4 / 52

12. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Outline
1
Stability of Linear ODEs
Basic Idea
Stability Conditions
Examples
2
Linear Stability Analysis
General Theory
Examples
3
Phase Plane and Stability Analysis
Phase Portraits
Examples
4
Lyapunov Stability
Stability i.s.L.
Lyapunov's Stability Theorem
Variable Gradient Method
Global Asymptotic Stability
Lyapunov's Indirect Method
5 / 52

13. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability of Linear ODEs
n
th
-order ODE
anx
(n)
+an1x
(n1)
+: : :+a1x
0
+a0x=f(t)
2
nd
-order ODE
a2x
00
+a1x
0
+a0x=f(t)
6 / 52

14. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability of Linear ODEs
n
th
-order ODE
anx
(n)
+an1x
(n1)
+: : :+a1x
0
+a0x=f(t)
2
nd
-order ODE
a2x
00
+a1x
0
+a0x=f(t)
6 / 52

15. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Basic Idea
ODEa2x
00
+a1x
0
+a0x=f(t)
General solutionx=c1x1+c2x2+xpwherec1,c2are
arbitrary constants andxpis a particular solution.
The system is stable
8c1;c2
() c1x1+c2x2!0
as t! 1
7 / 52

16. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Basic Idea
ODEa2x
00
+a1x
0
+a0x=f(t)
General solutionx=c1x1+c2x2+xpwherec1,c2are
arbitrary constants andxpis a particular solution.
The system is stable
8c1;c2
() c1x1+c2x2!0
as t! 1
7 / 52

17. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Basic Idea
ODEa2x
00
+a1x
0
+a0x=f(t)
General solutionx=c1x1+c2x2+xpwherec1,c2are
arbitrary constants andxpis a particular solution.
The system is stable
8c1;c2
() c1x1+c2x2!0
as t! 1
7 / 52

18. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability Conditions
ODE a2x
00
+a1x
0
+a0x=f(t)
Characteristic equationa2m
2
+a1m+a0= 0
Table:
roots solution stability condition
m16=m2 c1e
m1t
+c2e
m2t
m1;m2<0
m=m1=m2e
mt
(c1+c2t) m<0
m1;2=i e
t
(c1cost+c2sint) <0
Summary stableif and only if all the
rootsof the characteristic equation have
negative real parts.
8 / 52

19. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability Conditions
ODE a2x
00
+a1x
0
+a0x=f(t)
Characteristic equationa2m
2
+a1m+a0= 0
Table:
roots solution stability condition
m16=m2 c1e
m1t
+c2e
m2t
m1;m2<0
m=m1=m2e
mt
(c1+c2t) m<0
m1;2=i e
t
(c1cost+c2sint) <0
Summary stableif and only if all the
rootsof the characteristic equation have
negative real parts.
8 / 52

20. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability Conditions
ODE a2x
00
+a1x
0
+a0x=f(t)
Characteristic equationa2m
2
+a1m+a0= 0
Table:
roots solution stability condition
m16=m2 c1e
m1t
+c2e
m2t
m1;m2<0
m=m1=m2e
mt
(c1+c2t) m<0
m1;2=i e
t
(c1cost+c2sint) <0
Summary stableif and only if all the
rootsof the characteristic equation have
negative real parts.
8 / 52

21. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 1.1
ODEx
00
+ 11x
0
+ 24x= 33t
Characteristic equationm
2
+ 11m+ 24 = 0Rootsm=8<0 m=3<0The ODE isstable.
9 / 52

22. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 1.1
ODEx
00
+ 11x
0
+ 24x= 33t
Characteristic equationm
2
+ 11m+ 24 = 0Rootsm=8<0 m=3<0The ODE isstable.
9 / 52

23. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 1.1
ODEx
00
+ 11x
0
+ 24x= 33t
Characteristic equationm
2
+ 11m+ 24 = 0Rootsm=8<0 m=3<0The ODE isstable.
9 / 52

24. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 1.1
ODEx
00
+ 11x
0
+ 24x= 33t
Characteristic equationm
2
+ 11m+ 24 = 0Rootsm=8<0 m=3<0The ODE isstable.
9 / 52

25. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 1.2
ODEx
00
8x
0
+ 17x= 0
Its characteristic equation is
m
2
8m+ 17 = 0
Its roots are
m1;2= 4i
HereRe(m1;2) = 4>0. Hence the dierential
equation isunstable.
10 / 52

26. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 1.2
ODEx
00
8x
0
+ 17x= 0
Its characteristic equation is
m
2
8m+ 17 = 0
Its roots are
m1;2= 4i
HereRe(m1;2) = 4>0. Hence the dierential
equation isunstable.
10 / 52

27. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
Autonomous system_x=f(x)
Equilibrium pointx

, thenf(x

) =0.Multivariate Taylor expansion
_x=f(x

) +fx(x

)(xx

) +: : :
=fx(x

)(xx

) +: : :
Ifx= (x1;x2; : : : ;xn) andf= (f1;f2; : : : ;fn), then
the partial derivativefx(x

) can be interpreted as
theJacobianevaluated atx

,J

=J(x

) =
@f
@x
.
11 / 52

28. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
Autonomous system_x=f(x)
Equilibrium pointx

, thenf(x

) =0.Multivariate Taylor expansion
_x=f(x

) +fx(x

)(xx

) +: : :
=fx(x

)(xx

) +: : :
Ifx= (x1;x2; : : : ;xn) andf= (f1;f2; : : : ;fn), then
the partial derivativefx(x

) can be interpreted as
theJacobianevaluated atx

,J

=J(x

) =
@f
@x
.
11 / 52

29. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
Autonomous system_x=f(x)
Equilibrium pointx

, thenf(x

) =0.Multivariate Taylor expansion
_x=f(x

) +fx(x

)(xx

) +: : :
=fx(x

)(xx

) +: : :
Ifx= (x1;x2; : : : ;xn) andf= (f1;f2; : : : ;fn), then
the partial derivativefx(x

) can be interpreted as
theJacobianevaluated atx

,J

=J(x

) =
@f
@x
.
11 / 52

30. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
Autonomous system_x=f(x)
Equilibrium pointx

, thenf(x

) =0.Multivariate Taylor expansion
_x=f(x

) +fx(x

)(xx

) +: : :
=fx(x

)(xx

) +: : :
Ifx= (x1;x2; : : : ;xn) andf= (f1;f2; : : : ;fn), then
the partial derivativefx(x

) can be interpreted as
theJacobianevaluated atx

,J

=J(x

) =
@f
@x
.
11 / 52

31. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
Taylor expansion
_x=f(x

) +fx(x

)(xx

) +: : :
=fx(x

)(xx

) +: : :
Takingx=xx

implies_x=_x.Ifxis very small, then
_x=J

x
12 / 52

32. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
Taylor expansion
_x=f(x

) +fx(x

)(xx

) +: : :
=fx(x

)(xx

) +: : :
Takingx=xx

implies_x=_x.Ifxis very small, then
_x=J

x
12 / 52

33. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
Taylor expansion
_x=f(x

) +fx(x

)(xx

) +: : :
=fx(x

)(xx

) +: : :
Takingx=xx

implies_x=_x.Ifxis very small, then
_x=J

x
12 / 52

34. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
The equilibrium pointx

is
Stable J

have negative real
parts.
Unstable J

has
positive real part.
Oscillatory J

havezeroreal
parts andnonzeroimaginary parts.
13 / 52

35. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
The equilibrium pointx

is
Stable J

have negative real
parts.
Unstable J

has
positive real part.
Oscillatory J

havezeroreal
parts andnonzeroimaginary parts.
13 / 52

36. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Linear Stability Analysis
The equilibrium pointx

is
Stable J

have negative real
parts.
Unstable J

has
positive real part.
Oscillatory J

havezeroreal
parts andnonzeroimaginary parts.
13 / 52

37. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Table: Stability in relation to eigenvalues
14 / 52

38. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.1
System
_x1=4x1x2
_x2=2x1x2
Equilibrium point ;0).JacobianJ(0;0)=

41
21
!
Eigenvalues1=3<0 2=2<0
The system isstable.
15 / 52

39. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.1
System
_x1=4x1x2
_x2=2x1x2
Equilibrium point ;0).JacobianJ(0;0)=

41
21
!
Eigenvalues1=3<0 2=2<0
The system isstable.
15 / 52

40. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.1
System
_x1=4x1x2
_x2=2x1x2
Equilibrium point ;0).JacobianJ(0;0)=

41
21
!
Eigenvalues1=3<0 2=2<0
The system isstable.
15 / 52

41. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.1
System
_x1=4x1x2
_x2=2x1x2
Equilibrium point ;0).JacobianJ(0;0)=

41
21
!
Eigenvalues1=3<0 2=2<0
The system isstable.
15 / 52

42. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.1
System
_x1=4x1x2
_x2=2x1x2
Equilibrium point ;0).JacobianJ(0;0)=

41
21
!
Eigenvalues1=3<0 2=2<0
The system isstable.
15 / 52

43. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.1
_x1=4x1x2
_x2=2x1x2
16 / 52

44. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.2
Lotka-Volterra equations
_x1=ax1bx1x2
_x2=dx1x2cx2
Equilibrium points ;0) and (c=d;a=b).
JacobianJ(0;0)=

a0
0c
!
Eigenvalues1=a>0 and2=c<0.
(0;0) isunstable.
17 / 52

45. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.2
Lotka-Volterra equations
_x1=ax1bx1x2
_x2=dx1x2cx2
Equilibrium points ;0) and (c=d;a=b).
JacobianJ(0;0)=

a0
0c
!
Eigenvalues1=a>0 and2=c<0.
(0;0) isunstable.
17 / 52

46. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 2.2
JacobianJ(c=d;a=b)=

0bc=d
ad=b 0
!
Eigenvalues1;2=i
p
ac.
(c=d;a=b) isoscillatory.
18 / 52

47. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
What is a Phase Portrait?
powerful tool to represent the solutions of systems of
dierential equations in Cartesian plane.
drawn as parametric curves (with timetas the parameter). requires technological assists for accurate representation.
19 / 52

48. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
What is a Phase Portrait?
powerful tool to represent the solutions of systems of
dierential equations in Cartesian plane.
drawn as parametric curves (with timetas the parameter). requires technological assists for accurate representation.
19 / 52

49. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
What is a Phase Portrait?
powerful tool to represent the solutions of systems of
dierential equations in Cartesian plane.
drawn as parametric curves (with timetas the parameter). requires technological assists for accurate representation.
19 / 52

50. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
What is a Phase Portrait?
powerful tool to represent the solutions of systems of
dierential equations in Cartesian plane.
drawn as parametric curves (with timetas the parameter). requires technological assists for accurate representation.
19 / 52

51. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
General Theory
Homogeneous Linear System
x
0
=Ax
Equilibrium Point origin.
20 / 52

52. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Flows in relation to the eigenvalues ofA
21 / 52

53. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Flows in relation to the eigenvalues ofA
22 / 52

54. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Flows in relation to the eigenvalues ofA
23 / 52

55. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Flows in relation to the eigenvalues ofA
24 / 52

56. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.1
Linear system
x
0
1=2x1
x
0
2=x14x2
Eigenvalues1=2<0 and2=4<0.
Stable node. 25 / 52

57. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.1
Linear system
x
0
1=2x1
x
0
2=x14x2
Eigenvalues1=2<0 and2=4<0.
Stable node. 25 / 52

58. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.1
Linear system
x
0
1=2x1
x
0
2=x14x2
Eigenvalues1=2<0 and2=4<0.
Stable node. 25 / 52

59. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.2
Linear system
x
0
1=x1
x
0
2=x2
Eigenvalues1=1<0 and2= 1>0.
Saddle point (always unstable).
26 / 52

60. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.2
Linear system
x
0
1=x1
x
0
2=x2
Eigenvalues1=1<0 and2= 1>0.
Saddle point (always unstable).
26 / 52

61. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.3
Linear system
x
0
1= 3x2
x
0
2=x1
Eigenvalues1;2= 0
p
3i.
Equilibrium point is acenter. It isneutrally
stable.
27 / 52

62. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.4
Lotka-Volterra system
x
0
1= (10x2)x1
x
0
2= (10 +x1)x2
Equilibrium points ;0) and (10;10).
Eigenvalues of the linearized system ;0) are1;2=10.
at (10;10) are1;2=10i.
28 / 52

63. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.4
Lotka-Volterra system
x
0
1= (10x2)x1
x
0
2= (10 +x1)x2
Equilibrium points ;0) and (10;10).
Eigenvalues of the linearized system ;0) are1;2=10.
at (10;10) are1;2=10i.
28 / 52

64. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.4
Lotka-Volterra system
x
0
1= (10x2)x1
x
0
2= (10 +x1)x2
Equilibrium points ;0) and (10;10).
Eigenvalues of the linearized system ;0) are1;2=10.
at (10;10) are1;2=10i.
28 / 52

65. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 3.4
(0;0) is asaddle point.
(10;10) is acenter.
29 / 52

66. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Who was Lyapunov?
Figure:
Russian mathematician, mechanician and physicist.
known for his development of the stability theory of a
dynamical system.
doctoral thesisThe general problem of the stability of
motion, (1892).
30 / 52

67. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability in the sense of Lyapunov (i.s.L.)
Autonomous system_x=f(x)
Equilibrium pointx=0
Stableif, for each >0,9=()>0 such that
jjx(0)jj< =) jjx(t)jj< 8t0
31 / 52

68. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability in the sense of Lyapunov (i.s.L.)
Autonomous system_x=f(x)
Equilibrium pointx=0
Unstableif the stability conditions are violated.
32 / 52

69. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Stability in the sense of Lyapunov (i.s.L.)
Autonomous system_x=f(x)
Equilibrium pointx=0
Asymptotically stableif it is not only stable but also any
solution starting very close to it eventually converges to it,
i.e.,9 >0 such that
jjx(0)jj< =)lim
t!1
x(t) =0
33 / 52

70. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Lyapunov's Stability Theorem
DR
n
is a domain containingx=0andV:D!Ris a
C
1
s.t.V(0) = 0 and
V(x)>0in D f0g (1)
_V(x)0in D (2)
thenx=0isstable.
Also, if
_V(x)<0in D f0g (3)
thenx=0isasymptotically stable.
34 / 52

71. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Lyapunov's Stability Theorem
DR
n
is a domain containingx=0andV:D!Ris a
C
1
s.t.V(0) = 0 and
V(x)>0in D f0g (1)
_V(x)0in D (2)
thenx=0isstable.
Also, if
_V(x)<0in D f0g (3)
thenx=0isasymptotically stable.
34 / 52

72. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Geometric Interpretation
_V(x) =
nP
i=1
@V
@xi
fi(x) =
@V
@x
f(x)
_
V0 ensures that oncex(t) moves insideV(x) =c, it
can never come out again.
_V<0 ensures thatx(t) keeps moving inside, until it falls
into the origin.
35 / 52

73. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Geometric Interpretation
_V(x) =
nP
i=1
@V
@xi
fi(x) =
@V
@x
f(x)
_
V0 ensures that oncex(t) moves insideV(x) =c, it
can never come out again.
_V<0 ensures thatx(t) keeps moving inside, until it falls
into the origin.
35 / 52

74. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.1
Pendulum equation without friction
_x1=x2
_x2=
g
l
sinx1
Lyapunov function candidate
V(x) =
g
l
(1cosx1) +
1
2
x
2
2
V(0) = 0 andV(x) is positive denite over the
domainx12(2;2). Its derivative along the
solution trajectories is
_
V(x) = 0.
Thusx=0isstable. But it is not asymptotically
stable as_V(x) = 0.
36 / 52

75. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.1
Pendulum equation without friction
_x1=x2
_x2=
g
l
sinx1
Lyapunov function candidate
V(x) =
g
l
(1cosx1) +
1
2
x
2
2
V(0) = 0 andV(x) is positive denite over the
domainx12(2;2). Its derivative along the
solution trajectories is
_
V(x) = 0.
Thusx=0isstable. But it is not asymptotically
stable as_V(x) = 0.
36 / 52

76. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.1
Pendulum equation without friction
_x1=x2
_x2=
g
l
sinx1
Lyapunov function candidate
V(x) =
g
l
(1cosx1) +
1
2
x
2
2
V(0) = 0 andV(x) is positive denite over the
domainx12(2;2). Its derivative along the
solution trajectories is
_
V(x) = 0.
Thusx=0isstable. But it is not asymptotically
stable as_V(x) = 0.
36 / 52

77. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Finding Lyapunov function candidates
Possible ways:
using the total energy functions using quadratic forms employing theVariable Gradient Method
37 / 52

78. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Finding Lyapunov function candidates
Possible ways:
using the total energy functions using quadratic forms employing theVariable Gradient Method
37 / 52

79. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Finding Lyapunov function candidates
Possible ways:
using the total energy functions using quadratic forms employing theVariable Gradient Method
37 / 52

80. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Finding Lyapunov function candidates
Possible ways:
using the total energy functions using quadratic forms employing theVariable Gradient Method
37 / 52

81. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.2
Pendulum equation with friction
_x1=x2
_x2=
g
l
sinx1
k
m
x2
Lyapunov function candidate
V(x) =
g
l
(1cosx1) +
1
2
x
2
2
_V(0) = 0
Faulty conclusionx=0isonlystable.
38 / 52

82. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.2
Pendulum equation with friction
_x1=x2
_x2=
g
l
sinx1
k
m
x2
Lyapunov function candidate
V(x) =
g
l
(1cosx1) +
1
2
x
2
2
_V(0) = 0
Faulty conclusionx=0isonlystable.
38 / 52

83. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.2
Pendulum equation with friction
_x1=x2
_x2=
g
l
sinx1
k
m
x2
Lyapunov function candidate
V(x) =
g
l
(1cosx1) +
1
2
x
2
2
_V(0) = 0
Faulty conclusionx=0isonlystable.
38 / 52

84. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.2
Taking a generalquadratic forminstead of
1
2
x
2
2
,
V(x) =
1
2
x
T
Px+
g
l
(1cosx1)
=
1
2

x1x2


p11p12
p12p22
!
x1
x2
!
+
g
l
(1cosx1)
Derivative
_V(x) =
g
l
(1p22)x2sinx1
g
l
p12x1sinx1
+

p11p12
k
m

x1x2+

p12p22
k
m

x
2
2
39 / 52

85. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.2
Imposing some conditions to ensure_V(x)<0,
_V(x) =
1
2
g
l
k
m
x1sinx1
1
2
k
m
x
2
2
So in the domainD=fx2R
2
:jx1j< g,
V(x)>0 _V(x)<0
Thusx=0isasymptotically stable.
40 / 52

86. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Variable Gradient Method
Considering
g(x) =rV=

@V
@x

T
Then
_
V(x) =
@V
@x
f(x) =g
T
(x)f(x)
Goal:Choosingg(x) in such a way that simultaneously it
would be thegradientof a positive denite functionV(x) and
would make the derivative_V(x) to be negative denite.
We must have
@gi
@xj
=
@gj
@xi
8i;j= 1;2; : : : ;n
Then
V(x) =
xZ
0
g
T
(y)dy=
xZ
0
n
X
i=1
gi(y)dyi
41 / 52

87. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Variable Gradient Method
Considering
g(x) =rV=

@V
@x

T
Then
_
V(x) =
@V
@x
f(x) =g
T
(x)f(x)
Goal:Choosingg(x) in such a way that simultaneously it
would be thegradientof a positive denite functionV(x) and
would make the derivative_V(x) to be negative denite.
We must have
@gi
@xj
=
@gj
@xi
8i;j= 1;2; : : : ;n
Then
V(x) =
xZ
0
g
T
(y)dy=
xZ
0
n
X
i=1
gi(y)dyi
41 / 52

88. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Variable Gradient Method
Considering
g(x) =rV=

@V
@x

T
Then
_
V(x) =
@V
@x
f(x) =g
T
(x)f(x)
Goal:Choosingg(x) in such a way that simultaneously it
would be thegradientof a positive denite functionV(x) and
would make the derivative_V(x) to be negative denite.
We must have
@gi
@xj
=
@gj
@xi
8i;j= 1;2; : : : ;n
Then
V(x) =
xZ
0
g
T
(y)dy=
xZ
0
n
X
i=1
gi(y)dyi
41 / 52

89. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.3
System
_x1=
6x1
(1 +x
2
1
)
3
+ 2x2
_x2=2x1+
2x2
1 +x
2
1
Letting
g(x) =rV=

a11x1+a12x2
a21x1+a22x2
!
Takinga22=a11= 2 anda12=a21= 0,
_V(x) =g
T
(x)f(x)
=
12x
2
1
(1 +x
2
1
)
3

4x
2
2
1 +x
2
1
42 / 52

90. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.3
System
_x1=
6x1
(1 +x
2
1
)
3
+ 2x2
_x2=2x1+
2x2
1 +x
2
1
Letting
g(x) =rV=

a11x1+a12x2
a21x1+a22x2
!
Takinga22=a11= 2 anda12=a21= 0,
_V(x) =g
T
(x)f(x)
=
12x
2
1
(1 +x
2
1
)
3

4x
2
2
1 +x
2
1
42 / 52

91. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.3
System
_x1=
6x1
(1 +x
2
1
)
3
+ 2x2
_x2=2x1+
2x2
1 +x
2
1
Letting
g(x) =rV=

a11x1+a12x2
a21x1+a22x2
!
Takinga22=a11= 2 anda12=a21= 0,
_V(x) =g
T
(x)f(x)
=
12x
2
1
(1 +x
2
1
)
3

4x
2
2
1 +x
2
1
42 / 52

92. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.3
_
V(x) =g
T
(x)f(x)
=
12x
2
1
(1 +x
2
1
)
3

4x
2
2
1 +x
2
1
Here_V(x)<0. So solving forV(x) we get,
V(x) =x
2
1+x
2
2
ObviouslyV(x)>0. Thus the system isasymptotically
stable.
43 / 52

93. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.3
_
V(x) =g
T
(x)f(x)
=
12x
2
1
(1 +x
2
1
)
3

4x
2
2
1 +x
2
1
Here_V(x)<0. So solving forV(x) we get,
V(x) =x
2
1+x
2
2
ObviouslyV(x)>0. Thus the system isasymptotically
stable.
43 / 52

94. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Global Asymptotic Stability
Def:If there is an asymptotically stable equilibrium point
x=0with the property that, for any initial statex, the
solution trajectory(t;x) approachesx=0ast! 1,
disregarding how largejjxjjis, then the equilibrium point is
calledglobally asymptotically stableequilibrium point.
TheoremConsideringx=0to be an equilibrium point. Also
consideringV:R
n
!Rto be aC
1
function so that
V(0) = 0 andV(x)>0;8x6=0
jjxjj ! 1=)V(x)! 1
_
V(x)<0;8x6=0
thenx=0is globally asymptotically stable.
This equilibrium point isunique.
44 / 52

95. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Global Asymptotic Stability
Def:If there is an asymptotically stable equilibrium point
x=0with the property that, for any initial statex, the
solution trajectory(t;x) approachesx=0ast! 1,
disregarding how largejjxjjis, then the equilibrium point is
calledglobally asymptotically stableequilibrium point.
TheoremConsideringx=0to be an equilibrium point. Also
consideringV:R
n
!Rto be aC
1
function so that
V(0) = 0 andV(x)>0;8x6=0
jjxjj ! 1=)V(x)! 1
_
V(x)<0;8x6=0
thenx=0is globally asymptotically stable.
This equilibrium point isunique.
44 / 52

96. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Global Asymptotic Stability
Def:If there is an asymptotically stable equilibrium point
x=0with the property that, for any initial statex, the
solution trajectory(t;x) approachesx=0ast! 1,
disregarding how largejjxjjis, then the equilibrium point is
calledglobally asymptotically stableequilibrium point.
TheoremConsideringx=0to be an equilibrium point. Also
consideringV:R
n
!Rto be aC
1
function so that
V(0) = 0 andV(x)>0;8x6=0
jjxjj ! 1=)V(x)! 1
_
V(x)<0;8x6=0
thenx=0is globally asymptotically stable.
This equilibrium point isunique.
44 / 52

97. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Lyapunov's Indirect Method
Theorem:Let us consider that the followingnonlinear
systemhas the originx=0as its equilibrium point.
_x=f(x)
wheref:D!R
n
is aC
1
function andDcontainsx=0. Let
us dene the following Jacobian matrix.
A=
@f
@x




x=0
Then
ifRe <0 for all eigenvalues ofA,x=0is
asymptotically stable.
ifRe >0 for at least one eigenvalue ofA,x=0is
unstable.
45 / 52

98. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Lyapunov's Indirect Method
Begin withV(x) =x
T
Pxas the Lyapunov function candidate
wherePis a real symmetric positive denite matrix.
Then
_V(x) =x
T
P_x+_x
T
Px
=x
T
(PA+A
T
P)x
=x
T
Qx
where
PA+A
T
P=Q (4)
ThisQis a symmetric matrix.
IfQis positive denite, thenx=0isasymptotically stable.
46 / 52

99. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Lyapunov's Indirect Method
Begin withV(x) =x
T
Pxas the Lyapunov function candidate
wherePis a real symmetric positive denite matrix.
Then
_V(x) =x
T
P_x+_x
T
Px
=x
T
(PA+A
T
P)x
=x
T
Qx
where
PA+A
T
P=Q (4)
ThisQis a symmetric matrix.
IfQis positive denite, thenx=0isasymptotically stable.
46 / 52

100. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Lyapunov's Indirect Method
Why \Indirect"?
Usually we start with a Lyapunov function candidate and then
check negative deniteness of its derivative.
But here we rst choose a real symmetric positive denite
matrixQand then solve forP, given that it is solvable.
IfPis positive denite, thenx=0isasymptotically stable.
47 / 52

101. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Lyapunov's Indirect Method
Why \Indirect"?
Usually we start with a Lyapunov function candidate and then
check negative deniteness of its derivative.
But here we rst choose a real symmetric positive denite
matrixQand then solve forP, given that it is solvable.
IfPis positive denite, thenx=0isasymptotically stable.
47 / 52

102. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Lyapunov's Indirect Method
Why \Indirect"?
Usually we start with a Lyapunov function candidate and then
check negative deniteness of its derivative.
But here we rst choose a real symmetric positive denite
matrixQand then solve forP, given that it is solvable.
IfPis positive denite, thenx=0isasymptotically stable.
47 / 52

103. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.4
Pendulum equation with friction
_x1=x2
_x2=
g
l
sinx1
k
m
x2
Equilibrium points ;0) and (;0).
Jacobian matrix
@f
@x
=

0 1

g
l
cosx1
k
m
!
48 / 52

104. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.4
Jacobian matrix at ;0) is
A=

0 1

g
l

k
m
!
For a particular system, we may haveA=

0 1
24
!
. Taking
Q=I2and solvingPA+A
T
P=QforP, we get
P=

1:18750:5
0:5 0:3750
!
with leading principal minors 1:1875>0 and 0:1953>0. So
(0;0) isasymptotically stable.
49 / 52

105. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Example 4.4
Jacobian matrix at ;0) is
A=

0 1
g
l

k
m
!
For a particular system, we may haveA=

0 1
24
!
. Taking
Q=I2and solvingPA+A
T
P=QforP, we get
P=

0:93750:5
0:50:1250
!
with leading principal minors0:9375<0 and0:1328<0.
So (;0) isunstable.
50 / 52

106. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Phase portrait of a particular pendulum equation
A particular pendulum equation
_x1=x2
_x2=2 sinx14x2
Equilibrium points ;0) and (;0).
51 / 52

107. Stability
Analysis of
Dierential
Equations
Stability of
Linear ODEs
Basic Idea
Stability Conditions
Examples
Linear
Stability
Analysis
General Theory
Examples
Phase Plane
and Stability
Analysis
Phase Portraits
Examples
Lyapunov
Stability
Stability i.s.L.
Lyapunov's Stability
Theorem
Variable Gradient
Method
Global Asymptotic
Stability
Lyapunov's Indirect
Method
Thank You!
52 / 52