Recent content by Rubik

Homework Statement Prove carefully that the following map has a horseshoe for one positive value of \mu xn+1 = xn2 - \mu For what positive values of \mu will the map have a horseshoe? Homework Equations The Attempt at a Solution xn+1 = xn2 - \mu = g(xn,\mu) ∂g/∂x = 2x → turning pt...

Homework Statement If β=0 the neurone model is \dot{u}= -u \dot{v}= v2 + v - u + \delta If \delta = 1/4 it has critical point (0,-1/2) Transform the system so that the critical point is at the origin so let \bar{v} = v +1/2 and find the equations of motion for (u,\bar{v}) Homework...

No problem thanks so much for your help!

Yes it did, but I didn't realize there was a second part that asks to determine its stability characteristics..

Suppose \alpha(x0) enters and does not leave some closed and bounded domain D that contains no critical points. This means that \phi(x0, t) \in D for all t≥\tau, for some \tau≥0. Then there is at least one periodic orbit in D and this orbit is in the \omega-limit set of x0. What does this...

So the flow enters the trapping region about the origin which has no critical points which means it is a stable periodic orbit in the trapping region..

There can't be any fixed points in trapping region. There can be a fixed point at the origin? So to determing the stability I determine the fixed point at the origin?

So if I have to give bounds for the periodic orbit I can pick any value for r(small) provided r'>0 and then choose another value (this time larger) and show r'<0? How would I then determine its stability characteristics?

Okay so does this mean 1≤g(θ)≤3 So you get \dot{r} = 2r - r3g(θ) \dot{r} > 2r - 3r3 > 0 and so 2/3 > r2 \dot{r} < 2r - r3 < 0 and so r2 > 2 and √(2/3) < r < √2

Homework Statement System in polar coordinates \dot{r} = 2r - r3(2 + sin(\theta)), \dot{\theta} = 3 - r2 Use a trapping region to show there is at least one periodic orbit? Homework Equations By using Poincare Bendixson's Theorem The Attempt at a Solution I am struggling to...

Homework Statement z→x3+ i(1 - y)3: Show where the functions is analytic and differentiable. Homework Equations The Attempt at a Solution For a function to be analytic cauchy-riemann equations must hold.. so ux = vy and uy = -vx Now f(z) = x3 + i(1 - y)3 is already in the...

Homework Statement Identify the stable, unstable and center eigenspaces for \dot{y} = the 3x3 matrix row 1: 0, -3, 0 row 2: 3, 0, 0 row 3: 0, 0, 1 Homework Equations The Attempt at a Solution This is an example used from the lecture and I understand how to get the...

I have come up with this Taking the limit along the Real axis: lim as z→0 of (z/\bar{z})2 = lim (x + 0i)2/(x - 0i)2 = lim x2/x2 = 1 Then taking the limit at the points x + xi for x→0: lim as z→0 of (z/\bar{z})2 = lim (x + xi)2/(x - xi)2 = lim (2x2)/(-2x2) = -1 and since 1 ≠ -1...

Homework Statement Show that the lim z→0 of (z/\bar{z})2 does not exist Homework Equations The Attempt at a Solution Not to sure how to go about this question?

Homework Statement Find all complex solutions to \bar{z} = z Homework Equations z = x + iy and \bar{z} = x - iy The Attempt at a Solution What does it mean by find all complex solutions? \bar{z} = z 0 = x + iy - x + iy 0 = 2iy