MATH 2421 Introduction To Differential Equation

MATH 2421 Introduction To Differential Equation

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MATH 2421 Introduction To Differential Equation

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MATH 2421 Introduction To Differential Equation

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Course Code: MATH2421
University: Douglas College

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Country: Canada

Question:
Write the Introduction to Differential Equation For Positive Definite.
Answer:
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’ =   2×1 2ax2  -x1-x1x22
-2×2-2x12x2

=2×1(-x1-x12)+2ax2(-2×2-2x12x2)
=-2×12+(-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<-x2  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

b) x1’=x2

    x2’=-x1-x23 
 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’= 2×1  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’=-2×24 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. 

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’=   2×1  2ax2     -x2-x1(1-x12-x22)
X1-x2(1-x12-x22)
Multiplying this matrix we obtain
v’ =      -2×12+2×14-2ax22+2ax24+(-2+2a)x1x2+(2+2a)x12x22 
If we choose a=1 so that we eliminate the cross term we have 
v’ =  -2×12+2×14-2×22+2×24+4x12x22 
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 00 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.

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.

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

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 00
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))2×2
≤-(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=1/4×4-1/6×6 or       
 x-1/4×4+1/6×6=0  if we solve for the values of x we realize that they are bounded on .
8)x1’=-6×1/u2+2×2)
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’=   2×1(1+x12)-x12(2×1)/(1+x12)2    2×2  -6×1/1+x12+2×2 
=-12×12+4x1x2+4×13/(1+x12)3-4x1x2-4×22/1+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 Free Membership to World's Largest Sample Bank To View this & another 50000+ free samples. Please put your valid email id. E-mail Yes, alert me for offers and important updates Submit  Download Sample Now Earn back the money you have spent on the downloaded sample by uploading a unique assignment/study material/research material you have. After we assess the authenticity of the uploaded content, you will get 100% money back in your wallet within 7 days. UploadUnique Document DocumentUnder Evaluation Get Moneyinto Your Wallet Total 8 pages PAY 5 USD TO DOWNLOAD *The content must not be available online or in our existing Database to qualify as unique. Cite This Work To export a reference to this article please select a referencing stye below: APA MLA Harvard OSCOLA Vancouver My Assignment Help. (2021). MATH 2421 Introduction To Differential Equation. Retrieved from https://myassignmenthelp.com/free-samples/math2421-introduction-to-differential-equation/lyapunov-function.html. "MATH 2421 Introduction To Differential Equation." My Assignment Help, 2021, https://myassignmenthelp.com/free-samples/math2421-introduction-to-differential-equation/lyapunov-function.html. My Assignment Help (2021) MATH 2421 Introduction To Differential Equation [Online]. Available from: https://myassignmenthelp.com/free-samples/math2421-introduction-to-differential-equation/lyapunov-function.html[Accessed 18 December 2021]. My Assignment Help. 'MATH 2421 Introduction To Differential Equation' (My Assignment Help, 2021) accessed 18 December 2021.

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