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Dynamical systems questions
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Questions in this quiz (24)
1Classify the fixed point of the two-dimensional dynamical system:
- AIt is a saddle point.
- BIt is a stable spiral.
- CIt is an unstable spiral.
- DIt is a stable node
- EIt is an unstable node.
2The following system has a single fixed point:
Which of the following are stable directions of the fixed point?
- A(0,1)
- B(1,1)
- C(-1,1)
- D(-1,3)
- E(2,3)
3The normal forms of typical bifurcations for dynamical systems of dimensionality one are the following:
How does the stability time of the fixed points close to a transcritical bifurcation scale with the bifurcation parameter r?
- A~1/r
- B~1/sqrt(r)
- C~1
- D~sqrt(r)
- E~r
4Consider the dynamical system:
Which of the following choices of g(x, y) make the system conservative?
- Ax^2
- Be^(-x)
- Cy
- Dy^2
- Ex^2y
5The figure below shows the phase portrait of the system:
What is the sum of the indices of all fixed points of this system?
- A-2
- B-1
- C0
- D1
- E2
6Which of the following properties are satisfied in a continuous dynamical system with a globally attracting strange attractor?
- AClose-by trajectories remain close-by at all times
- BAt least one Lyapunov exponent must be negative
- CIf the dimensionality is two, the attractor formed in a stretch
and fold process - DThe Lyapunov dimension DL satisfies D_L ≤ dimensionality n
- EThe strange attractor forms a fractal structure
7Classify the fixed point of the two-dimensional dynamical system:
- AIt is a saddle point.
- BIt is a stable spiral.
- CIt is an unstable spiral.
- DIt is a stable node.
- EIt is an unstable node.
8The normal forms of typical bifurcations for dynamical systems of dimensionality one are the following:
How does the stability time of the fixed points close to a subcritical
pitchfork bifurcation scale with the bifurcation parameter r?- A∼1/r
- B∼1/sqrt(r)
- C∼1
- D∼sqrt(r)
- E∼r
9Which type(s) of bifurcation(s) does the system x˙= 5−re(−x^2) have?
- ASaddle-node bifurcation
- BTranscritical bifurcation
- CSupercritical pitchfork bifurcation
- DSubcritical pitchfork bifurcation
- ENo bifurcation occurs
10Each path A–E in the ∆-τ diagram below (∆ and τ are the determinant and trace of the stability matrix of dimensionality two) is obtained
from τ and ∆ of a fixed point upon increasing a parameter.
Which of the paths corresponds to one of the fixed points in a normal
form of the saddle-node bifurcation in dimensionality two?
- AA.
- BB.
- CC.
- DD.
- EE.
11The figure below shows the phase portrait of the system:
x˙ = y^3
y˙ = x
What is the index of the fixed point at the origin?
- A-2
- B-1
- C0
- D1
- E2
12Which properties hold in general for the Lyapunov exponents in a continuous dynamical system of dimensionality three with a strange attractor that is globally attracting?
- AOne Lyapunov exponent is negative
- BOne Lyapunov exponent is zero
- COne Lyapunov exponent is positive
- DThe sum of all Lyapunov exponents is negative
- EThe sum of all Lyapunov exponents is zero
13The fixed point of the following two-dimensional dynamical system has complex eigenvalues, select the correct statements.
- AThe fixed point is a stable spiral.
- BThe fixed point is a center.
- CThe fixed point is an unstable spiral.
- DTrajectories run clockwise (↻) in (x,y) space.
- ETrajectories run counterclockwise (↺) in (x,y) space.
14The figure below shows the phase portrait of the system:
x˙ = -y + xy
y˙ = x + y
What is the sum of the indices of all fixed points of this system?
- A-2
- B-1
- C0
- D1
- E2
15Which of the following statements about the indices of smooth flows of dimensionality two are correct?
- AThe index of a fixed point must either be -1 or 1.
- BThe index of a closed curve intersecting a fixed point is 0.
- CThe index of a closed curve notintersecting a fixed point is 1.
- DThe index of a closed orbit cannot be 0.
- EPitchfork bifurcations conserve the sum of fixed-point indices.
16Which of the following statements about bifurcations in smooth flows
are correct?- AA Hopf bifurcation is a local bifurcation.
- BA heteroclinic bifurcation is a local bifurcation.
- CThe period time of a limit cycle usually approaches infinity
close to a saddle-node bifurcation of limit cycles. - DIt is possible to have bifurcations between two unstable fixed
points, for example a saddle point and an unstable node. - EA homoclinic bifurcation usually occurs when a limit cycle
collides with a spiral.
17Which of the following statements about the generalized dimension spectrum D_q are true?
- AWhen q > 0 low-density regions of the attractor give the dominant contribution to D_q.
- BWhen q > 0 high-density regions of the attractor give the dominant contribution to D_q.
- CFor a monofractal, D_q is independent of q.
- DFor a multifractal, D_q decreases with increasing q.
- EThe Kaplan-Yorke conjecture states that, in most cases, the
Lyapunov dimension is equal to D_1.
18The figure below shows the first few generations in the construction of
a fractal. The fractal set is obtained by iterating to generation S_n with
n→∞.
Which of the following alternatives describe the box-counting dimension of the fractal above?
- Alog(2)/log(3)
- Blog(3)/log(4)
- Clog(4)/log(3)
- D3/2
- Elog(3)/log(2)
19Classify the fixed point of the two-dimensional dynamical system:
- AIt is a saddle point.
- BIt is a stable spiral.
- CIt is an unstable spiral.
- DIt is a stable node.
- EIt is an unstable node.
20Which of the following statements are true in general for smooth onedimensional dynamical systems (flows on the line)?
- AThey can be solved using separation of variables.
- BThey can have periodic solutions with finite period time.
- CThey can be chaotic.
- DAround any non-infinite initial position a unique solution exists within a non-empty time interval.
- EThe only possible non-infinite attractors are fixed points.
21Which of the following kinds of fixed points do you typically encounter in conservative dynamical systems
- ANodes
- BSaddles
- CSpirals
- DCenters
- EConservative systems do not have fixed points.
22A three-dimensional dynamical system has the following Lyapunov exponents: λ_1 = 1, λ_2 = 0, λ_3 = −1. Which of the following statements are true?
- AThe system may be chaotic.
- BThe system may be volume preserving.
- CThe system may be Hamiltonian.
- DThe system may have a globally attracting limit cycle.
- EThe Lyapunov spectrum is unchanged if time changes sign.
23The existence and uniqueness theorem implies that trajectories cannot intersect if the flow is smooth enough. But in many phase portraits of smooth systems different trajectories appear to intersect at fixed points, for example close to saddle points. Is this a contradiction? Answer with one of the following alternatives
- ANo, this is an artefact of projecting higher-dimensional trajectories onto two dimensions.
- BNo, since no trajectory can pass the fixed point, trajectories do not intersect.
- CNo, since flows are non-smooth at fixed points
- DNo, near the fixed point the density of trajectories become too high for the resolution of the phase portrait. Therefore trajectories appear to intersect even though they do not.
- ENo, it just appears that way due to numerical errors when plotting the phase portrait.
24The following sequence of images shows the phase portraits for the system
x' = r + x
y' = xy
for three values of r:
What is the sum of the indices of the two fixed points in the leftmost
panel with r = −1?
- A-2
- B-1
- C0
- D1
- E2
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- Dynamical system Quiz on Dynamical systems questions, created by David Larsson on 08/01/2020.
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- David Larsson
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