Non-Linear equations and more Flashcards
What are the properties of chaos?
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.
What is the definition of the similarity dimension?
ln(#copies) / ln(scale factor)
Give the terms in the Jacobian
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What is meant by homogenous solutions?
Steady in space (constant)
What are the additional properties of a strange attractor?
Compression of trajectories towards the attractor.
Divergence across the attractor.
Extreme sensitivity to initial conditions.
Often fractal properties.
Aperiodic.
What is the shape of a supercritical bifurcation?
What is its normal form?
Symmetry-breaking, sideways stable parabola, forwards pitchfork, x-axis is stable for negative x.
xdot = rx - x^3
What is the form of the logistic map?
Xn+1 = r * Xn * ( 1 - Xn )
What does a filled in circle mean in graphical analysis?
Stable fixed point.
What is the shape of a saddle-node bifurcation?
What is its normal form?
sideways parabola, top half is stable.
xdot = r - x^2
What is the shape of a subcritical bifurcation?
What is its normal form?
Backwards pitchfork, unstable parabola
xdot = rx + x^3
What is the definition of the compass dimension?
1 + ln(perimeter) / ln(1/ compass scale)
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?
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.
How would we define a potential?
What are its properties?
f(x) = - derivative of potential w.r.t. x
V always decreases along trajectories.
Local Vmax = unstable FP
Local Vmin = stable FP
What are the important differences to remember for LSA in discrete vs continuous systems?
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.
What are on the axes of a bifurcation diagram?
x against the parameter (r)
In what eigenvalue situation do we get borderline cases?
What determines whether we get a star node or a degenerate node?
Repeated eigenvalue.
If linearly independent eigenvectors can be chosen -> star
If not -> degenerate node (line)
What is the definition of a Lyapunov exponent?
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.
What is the equation for the eigenvalues of the jacobian in terms of the trace and determinant?
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What is the shape of an imperfect bifurcation?
What is its normal form?
(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
What is the definition of a limit cycle?
An attractor corresponding to a closed trajectory, which is not surrounded by other closed trajectories (unlike a centre).
How can we determine the stability of a cycle of period n?
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.
What is the general form of the solution to the jacobian equation for 2D LSA?
constant * e^(lambda t) * eigenvector
and then the second one
(lambda is the eigenvalue)
What is the shape of a transcritical bifurcation?
What is its normal form?
straight lines (both go through the origin, one is the x-axis) switch stability at the origin. xdot = rx - x^2
What is the meaning of a Lyapunov exponent?
chaotic attractors have positive exponents.
Stable FP’s and cycles have negative exponents.
What is the definition of the pointwise dimension?
What about the correlation dimension?
(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)
What are the properties of an attractor?
- 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)
reproduce the stability of 2D LSA diagram.
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What is a property of a “dissipative” map?
The determinant of the Jacobian is < 1.
What does “zero volume” mean?
In an n-dimensional space, an object of zero volume can be described by n-1 dimensions.
What is the definition of the box counting dimension?
limit for small boxes -> ln(#boxes filled) / ln(1/ box size)
What are the key features of a hopf bifurcation?
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.
What is a useful derivitave relation to use?
dy/dx = (dy/dt)/(dx/dt)
What is the formula for an infinite geometric series a * r^k?
a/(1-r)
(for magnitude of r < 1 (converges))
What is the quadratic equation?
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What is the equation for a taylor expansion of f(x) around the point x0?
the sum from n=0 of 1/n! (the nth derivitave of f evaluated at x0) (x-x0)^n
What is the euler method?
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