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Write the Introduction to Differential Equation For Positive Definite.

System 1: Global Asymptotically Stable Equilibrium Point

x=0 is the equilibrium point for this system. If we introduce a quadratic Lyapunov function

V=x12+ax22

Where a is a positive constant to be determined. v is positive definite on the entire state space ?2. V is also radially unbounded that is| v(x)|→∞ as? x?→∞. The derivative of v along the trajectories is given by 

V' =   2x1 2ax2  -x1-x1x22

-2x2-2x12x2

=2x1(-x1-x12)+2ax2(-2x2-2x12x2)

=-2x12+(-2-4a)x1x22-4ax22

If we choose a =-1/2 then we eliminate x1x2 and v' becomes 

v'=-2(x12-x22) .Therefore x=0 is globally asymptotically stable equilibrium point if x1>x2 or -x1<-x

since v(x) is a continuous differentiable function i.e.

v'(x)≤-k?x?a for all x??2 and where k, a>0

hence the origin is exponentially stable

  1. b) x1'=x2

    x2'=-x1-x2

 X=0 is the equilibrium point for this system. If we introduce a quadratic Lyapunov function

V=x12+ax22

Where a is a positive constant to be determined. v is positive definite on the entire state space ?2. V is also radially unbounded that is| v(x)|→∞ as? x?→∞. The derivative of v along the trajectories is given by 

V'= 2x1  2ax2    x2

-x1-x23     

 =2x1x2+2ax2(-x1-x23

=(2-2a)x1x2-2ax24)

If we choose a=1 we then eliminate the cross terms and we then have v'=-2x24 therefore x=0 is a globally asymptotically stable equilibrium point  

since v(x) is a continuous differentiable function i.e.

v'(x)≤-k?x?a for all x??2 and where k,a>0

hence the origin is globally exponentially stable. 

  1. c) x1'=-x2-x1(1-x12-x22)

     x2'=x1-x2(1-x12-x22)

 X=0 is the equilibrium point for this system. If we introduce a quadratic Lyapunov function

V=x12+ax22

Where a is a positive constant to be determined. v is positive definite on the entire state space ?2. V is also radially unbounded that is| v(x)|→∞ as? x?→∞. The derivative of v along the trajectories is given by      

V'=   2x1  2ax2     -x2-x1(1-x12-x22)

X1-x2(1-x12-x22)

Multiplying this matrix we obtain

v' =      -2x12+2x14-2ax22+2ax24+(-2+2a)x1x2+(2+2a)x12x2

If we choose a=1 so that we eliminate the cross term we have 

v' =  -2x12+2x14-2x22+2x24+4x12x2

Hence x=0 is not a globally asymptotically stable equilibrium point. since v(x) is a continuous differentiable function i.e.

v'(x)≥-k?x?a for all x??2 and where k, a>0

hence the origin is neither exponentially stable nor globally exponentially stable. 

2)  x'=σ(y-x)

     y'=rx-y-xz

     z'=xy-bz

where σ, r, b are positive constants and again we have that 0<r≤1

using the Lyapunov function

v(x,y,z)=x2/σ+y2+z2 where σ is to be determined

It is clear that v is positive definite on ?3 and is radially unbounded. The derivative of v along the trajectories of the system is given by 

v'=  2x/σ    2y    2z             σ(y-z)

rx-y-xz

xy-bz                                       

=-2x2-2y2-2bz2+2xy 2rxy

=-2[(x2+y2+bz2-(1+r)xy)]

=-2[(x-1/2(1+r)y)2+(1-(1+r/2)2)y2+1/2bz2)

Since 0<r≤1 it follows that 0<(1+r)/2<1 and therefore v' is negative definite on the entire state space ?3 hence the origin is globally asymptotically stable.

3.a) x1'=x2

X2'=-h1(x1)-x2-h2(x3)

X3'=x2-x3

Where h1 and h2 are locally Lipschitz function that satisfy hi(0)=0 and yhi(y)>0 for all y?0.

X2=0

  X3=0

h1(x1)-h2(0)=0

Since h2(0)=0 implies that h1(x1)=0 and therefore x1=0

Hence the system has a unique equilibrium point at the origin

b) v(x)=  1(y)dy +x22/22(y)dy

If we define v(x) the way it has been defined above, then it is clear that v(x) is locally positive definite.

  1. c) v'(x) =-h1(x1)2+x2-h2(x3)2 

Which is locally negative definite implying that x=0 or the origin is asymptotically stable equilibrium point.

  1. d) The function v(x) should be positive definite on the entire state space and has the property |v(x)|→∞ as ?x?→∞ and again

System 2: Globally Asymptotically Stable Equilibrium Point

v'(x) should be negative definite on the entire state space and this is only achieved by h1 and h2 satisfying hi(0)=0 and yhi(y)>0 for all y?0

4) x1'=x2

X2'=-sinx1-g(t)x2 

Where g(t) is continuously differentiable and satisfies 0<a≤g(t)≤β<∞,

g'(t)≤γ<2,for all t≥0

Lyapunov function

V(t,x)=1/2(asinx1 +x2)2 +[1 +ag(t) -a2](1-cosx1)

 To show that v(t,x) is positive definite and decrescent on Gr or origin. We assume that a>0

V(t,x)=1/2(asinx1+x2)2 +[1+ag(t)-a2]2(sin2x1/2)

Now ?2asin2x1/2?≤?x?2max(a/2,1/2) where a/2 condition in the max function arises from

|sinx1/2|≤|x1/2| for all x1

And 1/2 condition in the max function is for the case where x22/2 dominate if a is too small. Therefore w(?x?)=max(a/2,1/2)?x?2 is a class w function that bound v from above showing that v is decrescent on Gr.

To show that v is positive definite on Gr, we use

b) v'(t,x)=(asinx1+x2)(1+acosx1)x2+ag'(t)sinx1(-sinx1-g(t)x2)

=  a2x2/2sin2x1   +x2asinx1+ax2cosx1+x22-ag'(t)sin2 x1-a(g'(t))2x2

≤-(a-a)x22-a(2-γ)(1-cosx1)+0(|x|3) where 0(|x|3)is a term bounded by k|x|3 (k>0) in some neighbourhood of the origin.

c) From the above (b) we have shown that v'≤0 hence this suffices to show that the origin is uniformly asymptotically stable.

5) x1'=h(t)x2-g(t)x13

X2'=-h(t)x1-g(t)x23

X=0 is an equilibrium point of this system. If we introduce Lyapunov function  v(x)=x12+ax22 then we have

v'=2h(t)x12-2g(t)x14-2ah(t)x1x2-2ag(t)x24

if we choose a=0 then we have

v'=2(h(t)-g(t)x12)x12 and therefore v'≤0 if and only if g(t)>h(t) hence x=0 is uniformly asymptotically stable

since the Lyapunov function above is positive definite on the entire space such that |v(x)|→∞ as ?x?→∞ in addition v' is negative definite on the entire state space then x=0 which is the equilibrium point is globally uniformly asymptotically stable.

since v(x)  is a continuous differentiable function ie

v'(x)≤-k?x?a for all x??2 and where k, a>0

hence the origin is exponentially stable. If this happens in the entire state space then x=0 is said to be globally uniformly exponentially stable.

6)x'=x3-x5

0=x3-x5

X=1    hence x=0 is unstable equilibrium point.

To show that all solutions are bounded and defined on [0,∞) we integrate the equation so as to obtain

X=1/4x4-1/6x6 or       

 x-1/4x4+1/6x6=0  if we solve for the values of x we realize that they are bounded on .

8)x1'=-6x1/u2+2x2)

X2'=-2(x1+x2)/u2

Where u =1+x12  Let v(x)= x12/1+x12 +x22

=x12+x22+x12x22/u Therefore it is clear to see that v(x) >o for all x??2/{0} 

v'=   2x1(1+x12)-x12(2x1)/(1+x12)2    2x2  -6x1/1+x12+2x

=-12x12+4x1x2+4x13/(1+x12)3-4x1x2-4x22/1+x12

<-(4x1x2+4x22 /1+x12 which clearly shows that v' (x)<0 for all x??2/{0}

b) x2=2/x1-√2 The vector field on the boundary of this hyperbola, also the trajectories to the right of the branch in the first quadrant cannot cross that branch because the trajectories in the direction of the vector fields are approaching the boundary of ?2asymptotically and therefore they cannot cross them. 

c)This is simply because the hyperbola does not pass through the origin and for it to be globally asymptotically stable the function should be positive definite on the entire state space and in addition its derivative  should also be negative definite on the entire space that’s this conditions also apply to Lyapunov stability theorem and there it cannot contradict. 

References

V.  M.  ALEKSEEV, An  estimate  for  the  perturbations  of  the  solutions  of  ordinary  differen tial  equations,  Vestnik  Moskou.  Univ.  Ser. I.  Math.  Mekh,  2 (1961),  28-36.  [Russian]

Z.  S.  ATHANASSOV, Perturbation  theorems  for  nonlinear  systems  of  ordinary  differentialequations,  J.  Math.  Anal.  Appl.  86  (1982),  194-207.

I.  BIHARI,  A  generalization  of  a  lemma  of  Bellman  and  its  application  to  uniqueness problems  of  differential  equations,  Acta  Math.  Hungar.  7  (1956),  71-94.

G.  BIRKHOFF AND G.-C.  ROTA,  “Ordinary  Differential  Equations,”  3rd  ed.,  Wiley,  New York,  1978.

F.  BRAUER, Perturbations  of  nonlinear  systems  of  differential  equations,  J.  Math.  Anal. Appl.  14  (1967),  198-206.

F.  BRAUER, Perturbations  of  nonlinear  systems  of  differential  equations,  II,  J.  Mad  Anal. Appl.  17 (1967),  418434.

LIPSCHITZ  STABILITY  OF NONLINEAR  SYSTEMS 577 F. BRAUER  AND STRAUSS,Perturbations  of  nonlinear  systems  of  differential equations, III,  J.  Math.  Anal.  Appl. 31 (1970), 3748.

F.  BFCALJER, Perturbations  of  nonlinear  systems of  differential  equations, IV,  J. Math. Anal.  Appl.  31 (1972), 214-222

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