"Short questions" from old exams

Question 1

Give a definition for what a dynamical system is.

Answer 1

Dynamical system = Set of quantities (system) + Rule how these change with a single independent variable, usually time (dynamical)

Question 2

A nonautonomous system can be written as

xdot = f(x,t) ,

i.e. the flow f depends explicitly on time. Is a nonautonomous system a dynamical system? Explain your answer.

Answer 2

Yes, a nonautonomous system is a DS.

why though :/

Question 3

What does a transcritical bifurcation mean?

Answer 3

A transcritical bifurcation occurs when a fixed point exists both before and after the bifurcation, but it changes stability as r passes rc.

Question 4

What are the stable manifolds of a fixed point?

Answer 4

The stable manifold Ms of a fixed point
is either a point, curve, or surface in the phase-plane. It is defined as the set of points (including the fixed point) that approach the fixed point in the limit t →∞.

Question 5

Give an example of how the knowledge of stable manifolds of a fixed point could be used to understand the dynamics in a dynamical system.

Answer 5

The manifolds of FP often divide the phase space into regions of qualitatively different longtermdynamics (C.f. Strogatz 6.4).

(For example, a saddle point has one negative and one positive eigenvalue, it attracts along the stable direction, but repels along the unstable direction. Its stable and unstable manifolds are lines in these directions. Attractors|repellers are stable|unstable in all directions and the stable|unstable manifold is a surface (the entire phase plane). )

Question 6

What is a quasiperiodic flow? Give an example!

Answer 6

Quasiperiodicity is the property of a system that displays irregular periodicity. For example: Climate oscillations that appear to follow a regular pattern but which do not have a fixed period are called quasiperiodic

Question 7

In the problem sets the Lyapunov exponents were evaluated using a QR-decomposition method. Why is this method preferred over direct numerical evaluation of the eigenvalues of MTM where Mis the deformation matrix, or over evaluation of the Lyapunov exponent using separations between a number of particles?

Answer 7

The QR matrix decomposition allows us to compute the solution to the Least Squares problem.

Evaluation using the deformation matrix -> is given hard
numerics using separations -> Often works but is unreliable and complicated

Question 8

Sketch the typical shape of the generalized dimension spectrum Dq against q for a mono fractal and for a multi fractal.

Answer 8

Typical shape of mono fractal is a horizontal line and multi fractal as:
-arctan(q)
Changes sign at q=0

Question 9

What are the unstable manifolds of a fixed point?

Answer 9

The unstable manifold Mu consists
of the set of points that approach the fixed point in the limit t →−∞,
i.e. if the flow is reversed (t →−t), then Ms and Mu switch stability.

Question 10

What defines a conservative dynamical system?

Answer 10

• Frictionless dynamics
• Systems with at least one conserved quantity (integral of motion) are called conservative systems.

Question 11

What is the difference between a conservative dynamical system and a Hamiltonian dynamical system?

Answer 11

Hamiltonian phase space is an even dimensional space with a natural splitting into two sets of coordinates. It can be conservative if it not depend explicitly on time, since systems with at least one conserved quantity (integral of motion) are called conservative systems.

Question 12

Explain the main differences between a supercritical and a subcritical bifurcation.

Answer 12

Supercritical:
a stable fixed point becomes unstable at the bifurcation point, but two other stable fixed points are generated

Subcritical:
a stable fixed point becomes unstable at the bifurcation point, and two branches of stable fixed points are combined and eliminate eachother (3 fixed points to 1)

Question 13

Explain what a Hopf bifurcation is.

Answer 13

Critical point where a system’s stability switches and periodic solution arises

Question 14

State three properties of the index of a curve, IC .

Answer 14

• Index of a closed orbit Ic=1
• C– >C’ not passing a FP gives I_C=I_C’
• Reversing all arrows t –>-t leaves the index unchanged

Question 15

Explain what a fractal (strange) attractor is.

Answer 15

A minimal, attracting, invariant set that is aperiodic with chaotic dynamics. Strange attractors show self-similar structure at arbitrary small scales. These structures can be quantified using a fractal (non-integer) dimension D.

Question 16

What conditions must be satisfied for a system to show a fractal (strange) attractor?

Answer 16

*It is bounded but aperiodic (periodic would imply limit cycle).
*It requires a phase-space dimensionality of n>3. For n < 3, trajectories cannot pass and aperiodic motion is ruled out by the Poincar e-Bendixson theorem (Lecture 5).

Question 17

What is the significance of the parameter q in the generalized dimension spectrum Dq?

Answer 17

The significance of q can be summarized as:
* If q > 0 contributions to I(q, ) from regions of high density on the attractor are amplified compared to low-density regions.
Dq with large q therefore characterises clustering of high-density regions.
* When q < 0 the opposite is true: low-density regions dominate contributions to I(q, ) and Dq
* When q = 0 density variations are neglected. In this limit we recover the box-counting dimension (Eq. (1)): D0

Question 18

Explain a method that can be used to decrease the dimensionality of a dynamical system. What assumptions does your method rely on?

Answer 18

• Find conservation laws, which implies that some combination of phase-space variables is independent of time (a conserved quantity a.k.a. integral of motion) ⇒ One of the constituting variables can be eliminated, reducing the problem dimensionality.
• Use symmetries to decouple variables we are not interested in. For example, spherical symmetry allows to write the time evolution of the radial coordinate independently from the angular coordinates ⇒ dimensionality reduces from n = 3 to n = 1.
• Taking snapshots of a continuous dynamical system when its trajectory intersects a chosen lower-dimensional subspace (Poincaré map) result in a discrete system of lower dimensionality.

Question 19

Explain what it means that a dynamical system shows hysteresis. Give an example of a system with hysteresis.

Answer 19

A dynamical system shows hysteresis….

The problem of hysteresis could be catastrophic for ecological systems: if the system makes a big jump to a new equilibrium (for example due to human influence), it may be very hard to restore the system to its original state due to hysteresis.

Question 20

Give two examples of different types of global bifurcations. Explain how one can experimentally distinguish the different bifurcations.

Answer 20

Examples: Bifurcation between limit cycles, infinite period bifurcation, homoclinic bifurcation. Close to the bifurcation, all of these have characteristic dependence of the period time on the bifurcation parameter. Measuring how the period time changes with the bifurcation parameter one can distinguish the three types of bifurcations.

Question 21

In a dynamical system of dimensionality larger than two, what does the second largest Lyapunov exponent characterise?

Answer 21

It characterise if it’s volume conserving or dissipative

Question 22

Which kinds of fixed points do you typically encounter in conservative dynamical systems

Answer 22

Saddles and centers

Question 23

What is meant by a ghost or slow passage in the context of fixed points?

Answer 23

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).

Question 24

Explain what a limit cycle is. Give an example of a system with a limit cycle.

Answer 24

A 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.

Question 25

What is meant by structural stability?

Answer 25

Structural stability means that the topology of the flow does not change as the vector field is weakly perturbed. For example, linear centers aren’t structurally stable since the flow surrounding them may become either attracting or repelling under a small non-linear perturbation.

Question 26

Contrast any similarities or differences in calculating Lyapunov exponents for a continuous dynamical system (the Lorenz equations) and for a discrete dynamical system (the Hénon map).

Answer 26

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.

Question 27

A simple harmonic oscillator m*x’’=-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.

Answer 27

Since m*x’’=-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.

Question 28

What is meant by a catastrophe in the context of bifurcation theory?

Answer 28

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.

Question 29

What is meant by a reversible dynamical system? Give an example of a time reversible dynamical system of dimensionality two.

Answer 29

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.

Question 30

What does the Poincaré-Bendixon theorem state?

Answer 30

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.

Question 31

Explain the difference in the typical solutions of an integrable Hamiltonian system and a non-integrable Hamiltonian system.

Answer 31

In an integrable Hamiltonian system the solutions are periodi/quasiperiodic, while they are aperiodic, chaotic in a non-integrable system.

Question 32

Give a real-world example of a dynamical system with a transcritical bifurcation.

Answer 32

For example the logistic growth model xdot = rx(1 −x) has a transcritical bifurcation at r = 0. (Population, birth rate, death rate)

Question 33

Explain what the difference between a global and a local bifurcation is. Give two examples of global bifurcations.

Answer 33

• Local bifurcations, which can be analysed entirely through changes in the local stability properties of equilibria, periodic orbits or other invariant sets as parameters cross through critical thresholds; and
• Global bifurcations, which often occur when larger invariant sets of the system ‘collide’ with each other, or with equilibria of the system. They cannot be detected purely by a stability analysis of the equilibria (fixed points).

Example: Bifurcation between limit cycles, infinte-period bif., homoclinic bif.

Question 34

Explain what is meant by a secular term in perturbation theory.

Answer 34

When performing perturbation theory in a small parameter in a time dependent problem, it is common that the perturbation coefficients in grow without bound as t → ∞. Such terms growing without bound are denoted secular terms.

Question 35

What value does the maximal Lyapunov exponent of a stable limit cycle take? Explain why.

Answer 35

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).

Question 36

Explain why the transition from regular dynamics to chaos is typically very different in dissipative and in Hamiltonian dynamical systems.

Answer 36

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 transition instead occurs in bifurcations where closed orbits break up.

Question 37

What kind of fixed points do you typically encounter in Ham. sys?

Answer 37

Saddles and centers

Question 38

Explain a mechanism for obtaining self-sustained oscillations.

Answer 38

The typical example is an oscillator with negative damping in the linearized regime, resulting in an unstable spiral blowing up small oscillations. This blow up is counteracted by non-linear terms that regularize the oscillations at a finite amplitude. This is the mechanism underlying the van der Pol oscillator and many other limit cycles.

Question 39

Explain how the intermittency transition from regular dynamics to chaos (Pomeau-Manneville) works

Answer 39

A system with a stable regular attractor exhibit a bifurcation to a distant strange attractor (catastrophe). The ghost of the former fixed point affect the dynamics by introducing a bottleneck (slow region) where the dynamics is close to regular. After the trajectory escapes the bottleneck it moves around irregularly in phase space (intermittent chaotic outburst) until it reach the bottleneck again.

Question 40

Give an interpretation of the parameter q in Dq

Answer 40

Introduction of the parameter q weighs different boxes differently on the fractal set. When q=0, pk is taken to the power 0 in I(p,q),and all boxes are weighed equally. When q >0, large pk are weighed higher, meaning that the fractal dimension weigh dense regions of the fractal set higher. When q<0, sparse regions of the fractal set are weighed higher.

Question 41

Explain why the limit q->1 in Dq is not infinite.

Answer 41

Because pk is a probability it sums to unity. Hence I to 1 and ln to 0 which canceles.

Question 42

Explain what is meant by the Lyapunov time. Give examples of the approximate Lyapunov time for two systems.

Answer 42

The Lyapunov time is obtained by the inverse maximal Lyapunov exponent when it is positive (chaotic system). It determines the time scale under which the system is predictable. Two examples are the the solar system with Lyapunov time ~10^7 years, or weather systems with
Lyapunov time ~days

Question 43

Explain how the intermittency transition from regular dynamics to chaos (Pomeau-Manneville) works.

Answer 43

A system with a stable regular attractor exhibit a bifurcation to a distant strange attractor (catastrophe). The ghost of the former fixed point affect the dynamics by introducing a bottleneck (slow region) where the dynamics is close to regular. After the trajectory escapes the bottleneck it moves around irregularly in phase space (intermittent chaotic outburst) until it reach the bottleneck again.

Question 44

Explain what an integral of motion is.

Answer 44

An integral of motion is a quantity that is constant along a trajectory, but non-constant on open sets in phase space.

Question 45

Explain what libration and rotation is

Answer 45

On the cylinder periodic orbits comes in two types: librations and rotations. Librations always encircle fixed points whose indices sum to +1, while rotations instead encircle the cylinder.

Question 46

Explain what the difference between a limit set and an attractor is.

Answer 46

A limit set is given by the possible long-term dynamics of a system, for example fixed points, closed orbits or strange attractors. An attractor is a limit set that is stable to perturbations in all directions in phase space. For example, both bands of closed orbits and stable limit cycles are limit sets, but only the stable limit cycle is an attractor.

Question 47

Explain what transient chaos is. Give an example of a system with transient chaos.

Answer 47

Chaos is transient if the maximal Lyapunov exponent is positive at large but finite times, but negative in the long run. Examples: double pendulum, casting of dice, escape in three-body problem,

Question 48

Construct a non-linear dynamical system of dimensionality two with a non-linear center, i.e. a fixed point whose linear stability indicates a center and which is surrounded by closed orbits.

Answer 48

One example is the simple pendulum: x’‘=sin x

Question 49

What is required for a system to be volume conserving?

Answer 49

lambda1+lambda2+lambda3=0
equals to
Tr(J) = 0

Question 50

Name 3 things that are true about the generalized dimension
spectrum Dq.

Answer 50

• When evaluating Dq, the fraction of points in each point- containing box is weighted with a power q.
• The Kaplan-Yorke conjecture states that, in most cases, the information dimension is equal to D1.
• The correlation sum can be used to efficiently evaluate D2 for experimental data.

Question 51

Explain what is meant by critical slowing down after a saddle-node bifurcation. Give a real-world example.

Answer 51

At the bifurcation the flow velocity goes from zero to a small non-zero value. This causes the dynamics to slow down, causing it to appear to
be in an equilibrium, while it is in reality slowly leaving the apparent equilibrium. For example chemical reactions or after tipping points in
ecological/climate systems.

Question 52

Hamiltonian systems in one spatial dimension has centers at potential minima. What type(s) of fixed points can occur at minima of V pxqin the system 9x p and 9p V 1 pxq(note the sign)?

Answer 52

Saddle points

Question 53

Which bifurcations can be involved in an intermittency
transition to chaos?

Answer 53

• Saddle node bifurcation
• Hopf bifurcation
• Subcritical bifurcation

Question 54

Explain why the dynamics in billiard systems of rectangular shape is equivalent to the dynamics of uncoupled oscillators on a torus. What kinds of long-term behaviours are possible?

Answer 54

The dynamics on the torus is equivalent to a rectangular billiard if the trajectories are reflected along the centerline in both the x and y directions. The possible long-term behaviours in a rectangular billiard system are therefore the same as for the uncoupled oscillators on the torus, i.e. closed orbits or quasi-periodic trajectories.

Question 55

Explain why at least one Lyapunov exponent is zero for a trajectory starting on a closed orbit.

Answer 55

If one consider two initially closeby trajectories both starting on the
closed orbit, then they will end up at the same place every period time.
The initial separation vector between the two trajectories will therefore
neither grow or shrink in the long run, leading to a vanishing Lyapunov
exponent.

Question 56

Give an example of a physical system exhibiting self-sustained (limit cycle) oscillations.

Answer 56

Pacemaker

Question 57

Discuss why normal forms of bifurcations are useful in the context of dynamical systems.

Answer 57

The normal forms describe the universal forms of dynamical systems
close to bifurcations. They can therefore be used to analytically identify
bifurcations and their type, or to construct dynamical systems with
desired bifurcation properties.

Question 58

What form does Hamilton take?

Answer 58

x’ = dH/dp
p’ = -dH/dx