Non-Linear equations and more

Question 1

What are the properties of chaos?

Answer 1

In a deterministic system:

• aperiodic behaviour in the long term
• extreme sensitivity to changes in initial conditions

Need 3D for continuous systems, only 1D for discrete.

Question 2

What is the definition of the similarity dimension?

Answer 2

ln(#copies) / ln(scale factor)

Question 3

Give the terms in the Jacobian

Answer 3

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Question 4

What is meant by homogenous solutions?

Answer 4

Steady in space (constant)

Question 5

What are the additional properties of a strange attractor?

Answer 5

Compression of trajectories towards the attractor.
Divergence across the attractor.
Extreme sensitivity to initial conditions.
Often fractal properties.
Aperiodic.

Question 6

What is the shape of a supercritical bifurcation?
What is its normal form?

Answer 6

Symmetry-breaking, sideways stable parabola, forwards pitchfork, x-axis is stable for negative x.
xdot = rx - x^3

Question 7

What is the form of the logistic map?

Answer 7

Xn+1 = r * Xn * ( 1 - Xn )

Question 8

What does a filled in circle mean in graphical analysis?

Answer 8

Stable fixed point.

Question 9

What is the shape of a saddle-node bifurcation?
What is its normal form?

Answer 9

sideways parabola, top half is stable.
xdot = r - x^2

Question 10

What is the shape of a subcritical bifurcation?
What is its normal form?

Answer 10

Backwards pitchfork, unstable parabola
xdot = rx + x^3

Question 11

What is the definition of the compass dimension?

Answer 11

1 + ln(perimeter) / ln(1/ compass scale)

Question 12

How can we show that a classical system (observing a force that a potential can be defined for) conserves energy?
What are the properties of a conservative system?

Answer 12

Manipulate algebra to give the time derivative of the Hamiltonian as zero.
Trajectories are closed curves with constant energy.
There can be not attractive FP’s.
-> only centres and saddles.

Question 13

How would we define a potential?
What are its properties?

Answer 13

f(x) = - derivative of potential w.r.t. x
V always decreases along trajectories.
Local Vmax = unstable FP
Local Vmin = stable FP

Question 14

What are the important differences to remember for LSA in discrete vs continuous systems?

Answer 14

Discrete systems stable for lambda < 1 whereas it must be < 0 for continuous systems.
FP’s are where f(x) = 0 for continuous systems, but where f(x) = x for discrete.

Question 15

What are on the axes of a bifurcation diagram?

Answer 15

x against the parameter (r)

Question 16

In what eigenvalue situation do we get borderline cases?
What determines whether we get a star node or a degenerate node?

Answer 16

Repeated eigenvalue.
If linearly independent eigenvectors can be chosen -> star

If not -> degenerate node (line)

Question 17

What is the definition of a Lyapunov exponent?

Answer 17

The Lyapunov exponent at a point x is the limit as n tends to infinity of 1/n ln(the magnitude of the derivative of the nth iterate).
Only defined if it converges.

Question 18

What is the equation for the eigenvalues of the jacobian in terms of the trace and determinant?

Answer 18

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Question 19

What is the shape of an imperfect bifurcation?
What is its normal form?

Answer 19

(kind of like a supercritical) but depends on the pertubation delta.

if delta is positive (so taken away from xdot) the stability will continue on in the bottom quadrant).

xdot = rx - x^3 - delta

Question 20

What is the definition of a limit cycle?

Answer 20

An attractor corresponding to a closed trajectory, which is not surrounded by other closed trajectories (unlike a centre).

Question 21

How can we determine the stability of a cycle of period n?

Answer 21

The magnitude of the derivative of the nth iterate of the function. Equal to the magnitude of the product of the derivatives of the function at every point in the cycle.

Question 22

What is the general form of the solution to the jacobian equation for 2D LSA?

Answer 22

constant * e^(lambda t) * eigenvector
and then the second one
(lambda is the eigenvalue)

Question 23

What is the shape of a transcritical bifurcation?
What is its normal form?

Answer 23

straight lines (both go through the origin, one is the x-axis) switch stability at the origin. 
xdot = rx - x^2

Question 24

What is the meaning of a Lyapunov exponent?

Answer 24

chaotic attractors have positive exponents.
Stable FP’s and cycles have negative exponents.

Question 25

What is the definition of the pointwise dimension?
What about the correlation dimension?

Answer 25

(number of “points” within a circular region about point r) = (circle radius)^(the pointwise dimension)

(The average number of points over the whole structure) = (circle radius)^(the correlation dimension)

(useful for multifractals)

Question 26

What are the properties of an attractor?

Answer 26

• invariant set (trajectories starting in the attractor stay there)
• basin of attraction (all orbits enter the attractor in this region)
• minimal set (no proper subset)

Question 27

reproduce the stability of 2D LSA diagram.

Answer 27

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Question 28

What is a property of a “dissipative” map?

Answer 28

The determinant of the Jacobian is < 1.

Question 29

What does “zero volume” mean?

Answer 29

In an n-dimensional space, an object of zero volume can be described by n-1 dimensions.

Question 30

What is the definition of the box counting dimension?

Answer 30

limit for small boxes -> ln(#boxes filled) / ln(1/ box size)

Question 31

What are the key features of a hopf bifurcation?

Answer 31

2D system.
Stable FP at the origin until a>0.
For a>0 there is a limit cycle solution, which expands like a parabola for increasing a.

Question 32

What is a useful derivitave relation to use?

Answer 32

dy/dx = (dy/dt)/(dx/dt)

Question 33

What is the formula for an infinite geometric series a * r^k?

Answer 33

a/(1-r)

(for magnitude of r < 1 (converges))

Question 34

What is the quadratic equation?

Answer 34

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Question 35

What is the equation for a taylor expansion of f(x) around the point x0?

Answer 35

the sum from n=0 of 1/n! (the nth derivitave of f evaluated at x0) (x-x0)^n

Question 36

What is the euler method?

Answer 36

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