The text book for this assignment is available as pdf ingoogle. The text book is Differential Equations and Dynamical Systems, LawrencePerko , 3 rd edition, spring 2011. Document Preview: 1. The...

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The text book for this assignment is available as pdf in google. The text book is Differential Equations and Dynamical Systems, Lawrence Perko , 3rd
edition, spring 2011.




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1. The general form of a Lienard system is dx = y - f(x) dt dy =-x dt 3 If f (x)=µ(x - x) , this is the system form of Van der Pol’s equation. Find all the equilibria of the Van der Pol equation and characterize them as sources, sinks, etc. Make sure to analyze the influence of µ on the characterization. ________________________________ For more information on this application: Frequency demultiplication, Van der Pol, B. and van der Mark, Nature, 120, 363-364, (1927). A Lienard Oscillator Resonant Tunnelling Diode-Laser Diode Hybrid Integrated Circuit: Model and Experiment,, Slight, T.J.; Romeira, B.; Liquan Wang; Figueiredo, J.M.L.; Wasige, E.; Ironside, C.N. .; IEEE J. Quantum Electronics, Volume 44, Issue 12, Dec. 2008 Page(s):1158 – 1163.* 2. Komarova and co-workers have devised a simple 2-D dynamical system model of cell dynamics in bone remodeling. The equations associated with their model are: dx g g 1 11 21 =a x x -ß x 1 1 2 1 1 dt dx g g 2 12 22 =a x x -ß x 2 1 2 2 2 dt Here, we’re interested only in positive values of x and x , ß > 0 and ß > 0 , and the 1 2 1 2 parameters g may take on any real values. ij Find all equilibria (again, we’re only interested in the first quadrant) of the system. Identify degenerate cases. Find values of the parameters for which at least one of the equilibria is a source, other values for which at least one is a sink, others for which at least one is a saddle, others that yield a center, others that yield a stable focus, and others that yield an unstable focus. Sketch a phase diagram corresponding to each of these cases. Try to make some general statements about the structure of equilibrium as a function of the parameters. ________________________________ * Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling, S. V. Komarova, R. J. Smith, S. J. Dixon, S. M. Sims, L. M. Wahl, Bone...



Answered Same DayDec 23, 2021

Answer To: The text book for this assignment is available as pdf ingoogle. The text book is Differential...

Robert answered on Dec 23 2021
124 Votes
Problem 1
Answer
The given function is f(x) = μ(x3 – x)
Here we need to find the equilibrium points of the functions.
The given equations are,
So we can write it as,
x' = y - μ(x3
– x) ---(1)
y’ = -x ---(2)
For equilibrium points we need put the RHS term of the equation equal to zero.
y - μ(x3 – x) = 0
-x = 0
So, for x = 0 we get y = 0
From the first equation for x = , we can calculate values of y which will be zero. Hence there
will be three equilibrium points (0, 0), (1, 0), (-1, 0) for the system.
(0, 0) – Source
(1, 0) and (-1, 0) will be sink.
Influence of μ
Effect of μ is clearly visible on the limit cycles shown above for μ > 0. Shapes of the phase
diagrams are affected due to change in value of μ
Problem 2
Answer
For this case the given set of equation is,
To find equilibrium points we need to put the LHS term of the equation equal to zero. So we are
left with,
1x1
g11x2
g21 – β1x1 = 0
2x1
g12x2
g22 – β2x2 = 0
As per the question gij may take any values so, let g12 = g11 = g21 = g22 = 1 hence,
1x1x2 – β1x1 = 0
x1( 1x2 – β1) = 0
Here if x1 = 0 so, x2 = β1/ 1 so from the equation below we have,
2x1x2 – β2x2 = 0
x1 = β2/ 2
So the equilibrium points can be (0, β1/ 1) and (β2/ 2 , β1/ 1)
Degenerate cases can be when α1 or α2 will be zero.
For β1/ 1 = 1 equilibrium point (0,1) will be a source.
For β2/ 2 = 0 and β1/ 1 = -1 this will be a sink
And for saddle point both must be equal to each other or β1 1 , β2 = 2
It is clearly seen from the expressions of equilibrium points obtained that each equilibrium
point is a function of parameters only.
Problem 3
Answer
The given set of equations is,
To find out the equilibrium point we need to make left hand side part of the equation is zero so,
y = 0
And
x(0.5x2 -1 ) = 0
x =



So, the...
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