Nonlinear Dynamic SystemsHomework 2

1. Find the equilibrium solution(s) and bifurcation point(s) for the system system described byEq. 1. Next, classify the fixed points, the bifurcation type, and construct a bifurcation diagramusing β as a control parameter. Label stable and unstable solutions as well as the bifurcationpoint(s).

[x1

x2

]=

[x2

−2x2 − sin(x1) + β2

4 sin 2(x1)

]. (1)

2. Find the equilibrium solution(s) and bifurcation point(s) for the system system described byEq. 2. Next, classify the fixed points, the bifurcation type, and construct a bifurcation diagramusing β as a control parameter. Label stable and unstable solutions as well as the bifurcationpoint(s).

[x1

x2

]=

[x2

−2x2 + βx1 − x21

]. (2)

3. Aeroelastic galloping or flutter is a common problem that is encountered when specifying theflight envelope of an aerospace system (i.e. this constrains the flight speed of an aircraft). Findthe equilibrium solution(s) and bifurcation point(s) for the aeroelastic system described by Eq. 3[1]. Next, classify the fixed points, the bifurcation type, and construct a bifurcation diagram usingβ as a control parameter. Label stable and unstable solutions as well as the bifurcation point(s).Also, plot the response of the system, using the origin as an initial condition, for at least onepre-bifurcation and one post-bifurcation value. (see text p.77 for a hint).

[x1

x2

]=

[βx1 − ωx2 + (x1 − x2)(x2

1 + x22)

ωx1 + βx2 + (x1 + x2)(x21 + x2

2)

]. (3)

3. Quadratic damping is a common model for the drag force on a body moving through a fluidat high Reynolds numbers [1]. Assume a quadratic damping model for the pendulum of Fig. 1and derive the averaged solution for the pendulum free oscillations. Compare plots from numericalsimulation to the averaged solutions for parameters ζ = 0.05 and ω = 1 [rad/s] and initial conditionsθ = π/2 and θ = 0.

θ + ζ|θ|θ + ω2 sin θ = 0 . (4)

5. One of the first models to exhibit chaotic behavior in numerical simulation was a fluid convectionmodel proposed by Lorenz [2, 3]. This model is shown in Eq. (5), where x(t) represents a model

1

Pendulum

Pivot

Point

Figure 1: Schematic of a pendulum undergoing free vibrations.

of fluid velocity, y(t) is the temperature difference in the ascending and descending fluid, and z(t)temperature profile deviation from linearity [4]. Simulate the response of the system and plot x(t)vs. t, the three-dimensional phase space of the system, and the power spectrum (PSD). Simulatethe response over 100 time units with very small time steps starting from the initial conditionsx(0) = 0.3, y(0) = 0.5, z(0) = 0.6. Is the attractor a fixed point attractor, periodic attractor, orchaotic attractor. Make your justification based upon the requested three plots.

xyz

=σ(y − x)

rx − y − xz−bz + xy

. (5)

References

[1] A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics. New York, NY: John Wiley& Sons, Inc., 1 ed., 1995.

[2] F. C. Moon, Chaotic and Fractal Dynamics. Wiley Series in Nonlinear Science, New York, NY:John Wiley & Sons, Inc., 1 ed., 1992.

[3] E. N. Lorenz, “Deterministic non-periodic flow,” Journal of Atmospheric Science, vol. 20,pp. 130–141, 1963.

[4] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. United Kingdom: CambridgeUniversity Press, 2 ed., 2004.

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