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Dynamical systems Flashcards
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- 1Give a real-world example of a dynamical system with a transcritical bifurcation.For example the logistic growth model x˙ = rx(1−x) has a transcritical
bifurcation at r = 0. - 2Explain why the index IC of a curve C that is continuously transformed, without crossing any fixed points, into a new curve C 0 must remain constant, IC = IC0. Why may IC 6=IC0 if the curve passes a fixed point?Lecture notes 6.2
- 3Give a real-world example of a dynamical system with a transcritical bifurcation.For example the logistic growth model x˙ = rx(1−x) has a transcritical
bifurcation at r = 0. - 4Explain why the index I_C of a curve C that is continuously transformed, without crossing any fixed points, into a new curve C' must remain constant, I_C = I_C'. Why may I_C \ne I_C' if the curve passes a fixed point?Lecture notes 6.2
- 5Evaluate the nullclines and sketch the phase portrait in the upper right quadrant for the following dynamical system x˙ = x(2 − x − y) y˙ = y(x − 1)Nullclines:
x˙ = 0: x = 0, y = 2 − x
y˙ = 0: x = 1, y = 0
Phase portrait show two saddles at (0, 0) and (2, 0), and a stable spiral at (1, 1). - 6What is meant by a heteroclinic bifurcation? Let rc be the bifurcation point and sketch example phase portraits for r < r_c, r = r_c and r > r_c.Lecture notes 7.3.3
- 7Explain what a relaxation oscillator is and give a real-world example.Lecture notes 8.2
- 8Explain the difference between Lyapunov exponents and stability exponents of separations.Lecture notes 10
- 9Write down the equations for a Hamiltonian dynamical system of your choice.For example, starting from Newton’s second law of motion, F = mx¨,
we have the Hamiltonian system
x˙ = p/m
p˙ = F - 10What is meant by a catastrophe in the context of bifurcation theory?A catastrophe is a sudden change in the state of the system as a parameter is changed. For example, after a saddle-node bifurcation or a subcritical pitchfork bifurcation, the system quickly shifts to a distant attractor.
- 11Explain what the difference between a global and a local bifurcation is. Give two examples of global bifurcations.Bifurcations of cycles, infinite-period bifurcation, bifurcation of heteroclinic trajectories, bifurcation of homoclinic orbit, . . .
- 12Explain what is meant by a secular term in perturbation theory.When performing perturbation theory in a small parameter ε in a timedependent problem, it is common that the perturbation coefficients in ε grow without bound as t → ∞. Such terms growing without bound are denoted secular terms.
- 13What value does the maximal Lyapunov exponent of a stable limit cycle take?Explain why.The maximal Lyapunov exponent is zero. For a stable limit cycle closeby trajectories are attracted, meaning all Lyapunov exponents are negative, except for the Lyapunov exponent along the cycle which must be zero due to periodicity (separations can neither grow nor shrink in the long run because after one revolution of the limit cycle, the separation is back to the original length).
- 14Explain why the transition from regular dynamics to chaos is typically very different in dissipative and in Hamiltonian dynamical systems.In dissipative systems the transitions usually occurs though a sequence of bifurcations of where attractors becomes unstable. In Hamiltonian systems there are no attractors and the thransition instead occurs in bifurcations where closed orbits break up.
- 15A simple harmonic oscillator mx'' = -kx is a system that oscillates along the x-axis in one dimension. Explain how this is consistent with the statement that dynamical systems of dimensionality one cannot show oscillations.Since mx'' = -kx has a second derivative it can be written in terms of a dynamical system of dimensionality two, i.e. the statement about dimensionality one systems does not apply.
- 16What is meant by a catastrophe in the context of bifurcation theory?A catastrophe is a sudden change in the state of the system as a parameter is changed. For example, after a saddle-node bifurcation or a subcritical pitchfork bifurcation, the system quickly shifts to a distant attractor.
- 17What is meant by a reversible dynamical system? Give an example of a time reversible dynamical system of dimensionality two.A reversible system of dimensionality two is invariant under the simultaneous change t → -t and y → -y,
for example x' = y -y^3, y' = - x - y^2. - 18What does the Poincar´e-Bendixon theorem state?A trapping region is a closed region where the flow does not point outwards anywhere. The Poincaré-Bendixon theorem applies to dimensionality two systems. It states that for any trapping region that does not contain any fixed point, trajectories inside the trapping region must end up on a closed orbit.
- 19Construct a fractal set (for example a variation of the Cantor set) with fractal dimension D_0 = ln(3)/ln(5).One example is the even fifths Cantor set where intervals 2 and 4 are removed at each generation (out of 5 equal-sized intervals).
- 20Explain the difference in the typical solutions of an integrable Hamiltonian system and a non-integrable Hamiltonian system.In an integrable Hamiltonian system the solutions are periodi/quasiperiodic,
while they are aperiodic, chaotic in a non-integrable system. - 21Consider once again the system in Eq. (1) above. Is the index preserved in the bifurcation as r passes 0? Explain.No, the index is not preserved. For r < 0 the index is 2, after the bifurcation the index is 0. At the bifurcation point r = 0 the line x = 0 consists of non-isolated fixed points. Therefore index theorem does not apply when r = 0 and the index does not need to be conserved.
- 22What is meant by a ghost or slow passage in the context of fixed points?Close to for example a saddle-node bifurcation of a one-dimensional
dynamical system the magnitude of the flow is small where the fixed points used to be. This creates a bottleneck in the dynamics with slow passing time (typically of the order 1/√µ where µ is the bifurcation parameter of a bifurcation on normal form). - 23Explain what a limit cycle is. Give an example of a system with a limit cycleA limit cycle is a closed orbit that is isolated (at least from one side). It can be stable, unstable or half-stable. An example of a system with a limit cycle is the van der Pol oscillator.
- 24What is meant by structural stability?Structural stability means that the topology of the flow does not change
as the vector field is weakly perturbed. For example, linear centers are not structurally stable since the flow surrounding them may become either attracting or repelling under a small non-linear perturbation. - 25In the problem sets you calculated the Lyapunov exponents for a continuous dynamical system (the Lorenz equations) and for a discrete dynamical system (the Hénon map). Contrast any similarities or differences in the two approaches.For the continuous system a discretisation of the dynamics was used to calculate the deformation matrix M as a product of discrete matrices for short time intervals. Therefore the calculation of the Lyapunov exponents was basically identical to that of a discrete system, with the difference that the discrete matrix in the continuous case was given by I + Jδt, while it for the discrete system was given simply by J.
- 26—A suitable method is to plot ln(I(q, ε)) against ln(ε) to see in which range
we have a power-law scaling I(q, ε) ∼ A_qε^
((q−1)D_q). In this range the slope of the curve gives D_q independent from A_q:
D_q =1/(q − 1) ∆ln(I(q, ε))/∆ln(ε).
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- Dynamical system FlashCards sobre Dynamical systems Flashcards, criado por David Larsson em 08-01-2020.
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