1. Compute the largest lyapunov exponent for the map on the 2 dimensional plane x n+1 = where a is a real valued parameter
The limit defined by guarantees the validity of the linear approximation at any time of the operation.
Using the provided equation and matrix: x n+1 = , the highest value of a in the matrix is x n+1
Procedure
i. Starting with the initial condition in the basin of attraction, x n+1
ii. Iteration process until the orbit rests on the attractor
iii. Selecting the nearby point with a separation of d0
d= [(xa - xb)2 + (ya - yb)2]1/2
iv. Evaluation of the log |d1/d0|
We will then use base e for the calculation
2. Compute the largest lyapunov exponent for the map on the 2 dimensional plane - x n+1 = xn + where a is a real valued parameter
Generally, the largest Lyapunov exponent can be defined as shown below:
The limit defined by guarantees the validity of the linear approximation at any time of the operation.
The value of the highest Lyapunov exponent can therefore be estimated as
xb0 = xa1 + d0(xb1 - xa1) / d1 and yb0 = ya1 + d0(yb1 - ya1) / d1.
Whereby d is the separation.
3. True or false
a. Gradient dynamical systems can have asymptotically stable start nodes - False
b. Gradient dynamical systems can have saddle nodes - False
c. Gradient dynamical systems can undergo Hopf bifurcations as defined in class - True
4. The Lorenx equations are
Whereby R, and b are real valued parameters. Find condition(s) on these parameters that guarantee that these equations are volume contracting.
i. Non-linearity – in the equation, xy and xz are non-linearities
ii. The equations are invariant under (x,y)
(-x, -y). such that if (x(t); y(t); z(t)) is solution, then (-x(t), -y(t), -z(t)) is also a solution.
iii. The Lorenz equation is also dissipative whereby the volumes in phase-space contract under the flow.
iv. The values – (x*,y*,z*) = (0,0,0) should be a fixed point for all the values of the parameters. For the values r>1there should be a pair of fixed points. C at x* = y* = z* = r – 1. All these should coalesce with the origin as r 1+
5. Consider the system defined on the cylinder
a. Define the appropriate Poincare map, using the y axis as a Poincare section. Give a formula for the map
The Poincare map can be defined using the formula shown below. The successive intersections with a Poincare section can then be established using the formula shown below:
b. Show that the system has a periodic orbit
The system has a periodic orbit and can be shown using the formula ft+T(x) = ft(x) ; T > 0
The prime cycle p of the given period can on the other hand be said to be a single traversal of the periodic orbit. The shortest time Tp for which the equation ft+T(x) = ft(x) ; T > 0 has a solution can therefore be determined. A cycle point o a row ft which will cros the Poincar section np times is a fixed point of the Pn. Here there are a number of iterates in the poincar expression that needs to be established. All cycles here can therefore be referred to as fixed points.
c. Classify the stability of this periodic orbit for all the real values of the parameter a
The periodic orbit of all the real values of the parameter can be classified under
6. For the system
X= µx + x3, y = -y
7. Consider the map
= f (xn), f(x) = r – x2
a. Fixed points for all the positive values
In order to get the fixed points, we use fixed points iteration. If the function is defined in real numbers as the one shown above, the following sequences will be found: the values will then converge to a point x. therefore if the value of f is continuous, then the obtained value of x is a fixed point and the equation becomes. Therefore any positive value of x will become a fixed point.
b. Values of xc and rc
When computing the values, the following graphs are obtained and the values of xc and rc can be determined directly from the graph
The values of xc and rc can therefore be obtained as follows:
the first and second iterate (i.e., (Fc (x)) and (F 2c (x))) of the Ricker map when c=0.055 for different values of r. (b) Same as (a) except with c=0. (c) The fourth iterate F (4) c when r=3.5 with c=0 and c=0.055.
c) Value of x and r whereby the map has superstable fixed point
in order to get the value of x and r, the following equation applies:
xn+1=Fc (xn)=xn exp[r(1&x)]+c.
Using the above equation and substituting for the values of the x and y axes, the following graphs are obtained
Jordan, D. W.; Smith, P. (2007). Nonlinear Ordinary Differential Equations (fourth ed.). Oxford Univeresity Press.
Khalil, Hassan K. (2001). Nonlinear Systems. Prentice Hall.
Kreyszig, Erwin (1998). Advanced Engineering Mathematics. Wiley.
Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition. Springer.
