Questions and Answers of Fundamentals Of Vehicle Dynamics And Modelling

Fractals and Multifractals p p p3 42 p1 Figure 15.22: The motif used to construct the Koch curve multifractal, where p1 + p2 + p3 + p4 = 1.
A multifractal Koch curve is constructed and a weight is distributed as depicted in Figure 15.22, where p1 +p2 +p3 +p4 = 1. Determine αs and fs .
A multifractal square Koch curve is constructed and a weight is distributed as depicted in Figure 15.21. Plot the τ(q) curve and theDq and f (α) spectra when p1 = 1 9 and p2 = 1 3 .p p pp1 21 2p1
A multifractal Koch curve is constructed and the weight is distributed as depicted in Figure 15.20. Plot the f (α) spectrum when p1 = 1 3 and p2 = 1 6 .p p1 p2 2p1 Figure 15.20: The motif used to
Plot τ(q) curves and Dq and f (α) spectra for the multifractal Cantor set described in Example 1 when (i) p1 = 1 2 and p2 = 1 2 , (ii) p1 = 1 4 and p2 = 3 4 , and (iii) p1 = 2 5 and p2 = 3 5 .
Prove that D1 = lim l→0N i=1 pi ln(pi )−ln(l), by applying L’Hopital’s rule to equation (15.5).
Use the box-counting algorithm to approximate the fractal dimension of Barnsley’s fern. The Mathematica program for plotting the fern is given in Section 15.5.
ASierpi´nski square is constructed by removing a central square at each stage.Construct this fractal up to stage 3 and determine the fractal dimension of the theoretical object generated to infinity.
Consider Pascal’s triangle given below. Cover the odd numbers with small black discs (or shade the numbers). What do you notice about the pattern obtained?1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5
The Sierpi´nski triangle can be constructed by removing the central inverted equilateral triangle from an upright triangle; a motif is given in this chapter.Construct the Sierpi´nski triangle up to
(a) Consider the unit interval. A variation of the Cantor set is constructed by removing two line segments each of length 1 5 . Thus at stage 1, remove the segments between {1 5 . . . 2 5 } and {3 5
Apply the linear stability analysis to the iterative equation En+1 = i√1 − κEin + √κEn expi 2πn2Lλ0Aeff |En|2, for the parameter values given in this chapter. Compare the results with the
Consider the complex iterative equation En+1 = A + BEn exp[i(|En|2 − φL)], where B = 0.15 and φL represents a linear phase shift. Plot bifurcation diagrams for a maximum input intensity of A = 3
Plot the bifurcation diagram for the iterative equation in Exercise 6 for B = 0.15 when the input pulse is Gaussian with a maximum of 25Wm–2. How is the bistable region affected by the width of the
Plot bifurcation diagrams for Exercise 6, parts (a)–(c), when the maximum input intensity is 25Wm–2 and the input pulse is triangular.
Consider the complex iterative equation En+1 = A + BEn exp[i(|En|2)], used to model the SFR resonator. Use a linear stability analysis to determine the first bistable and unstable regions when (a) B
Consider the double-coupler nonlinear fiber ring resonator as shown in Figure 14.17.328 14. ElectromagneticWaves and Optical Resonators 3 : 1 − : 1 − L/2 L/2 E2 E 4 1 E ET Ein ER EFigure
Given the complex Ikeda mapping En+1 = A + BEn expiφ −C 1 + |En|2, where A, B, and C are constants, show that the steady-state solution, say, En+1 = En = ES, satisfies the equation cos C1 +
Given that En+1 = A + BEnei|En|2, prove that the inverse map is given by En+1 =En − A Bexp−i|En − A|2 B2.
Plot iterative maps for equation (14.8), using the parameter values given in the text, when κ = 0.0225 and (i) Ein = 4.5, (ii) Ein = 6.3, and (iii) Ein = 11.
Determine the number of fixed points of period one for system (14.10) when B = 0.4 and A = 3.9 by plotting the graphs of the simultaneous equations.
Plot the Mandelbrot set for the mapping zn+1 = z3n+ c.
Determine the periods of the points (i) c = −1.3 and (ii) c = −0.1 + 0.8i for the mapping zn+1 = z2n+ c.
Modify the Mathematica program in Section 13.3 to plot a Mandelbrot set for the mapping zn+1 = z4n+ c.Recommended Reading and Viewing 303
Determine the fixed points of periods one and two for the mapping zn+1 =z2n− 2zn + c.
Plot the Mandelbrot set for the mapping zn+1 = c − z2n.
Determine the boundaries of points of periods one and two for the mapping zn+1 = c − z2n.
Compute the fixed points of period one for the complex mapping zn+1 = 2 +znei|zn|2 10.
Consider equation (13.1); plot the Julia sets J(0, 0), J(−0.5, 0), J(−0.7, 0), and J(−2, 0).
Given that c = −1 + i, determine the fixed points of periods one and two for the mapping zn+1 = z2n+ c.
Consider the Julia set given in Figure 13.1(a). Take the mapping zn+1 =z2n+c, where c = −0.5 + 0.3i.(a) Iterate the initial point z0 = 0 + 0i for 500 iterations and list the final 100. Increase the
According toAhmed et al. [3], an inflation-unemployment model is given by Un+1 = Un−b(m−In), In+1 = In−(1−c)f (Un)+f (Un−b(m−In)), where f (U) = β1 +β2e−U, Un and In are measures of
(a) Consider the blood cell iterative equation (12.6). Assuming that b = 1.1 × 106, r = 8, and s = 16, show that there are (i) two stable and one unstable fixed points of period one when a = 0.2,
(a) Show that the Hénon map given by xn+1 = 1 − αx2 n + yn, yn+1 = βxn, where α > 0 and |β| < 1 undergoes a bifurcation from period-one to period-two behavior exactly when α = 3(β−1)2 4
Find the fixed points of periods one and two for the Hénon map given by xn+1 =3 50 +9 10 yn − x2 n, yn+1 = xn.Derive the inverse map.
Plot bifurcation diagrams for(a) the Gaussian map when α = 20 for −1 ≤ β ≤ 1;(b) the Gaussian map when β = −0.5 for 0 ≤ α ≤ 20.
Consider the iterative equation xn+1 = μxn(100 − xn), which may be used to model the population of a certain species of insect.Given that the population size periodically alternates between two
Consider the logistic map function defined by fμ(x) = μx(1−x). Determine the functions fμ(x), f 2μ(x), f 3μ(x), and f 4μ(x), and plot the graphs whenμ = 4.0. How many points are there of
By editing the Mathematica program given in Section 12.6, plot a bifurcation diagram for the tent map.
Nonlinear Discrete Dynamical Systems Sketch the graphs of T (x), T 2(x), and T 3(x). How many points are there of periods one, two, and three, respectively?(b) Let T be the function T : [0, 1] →
(a) Let T be the function T : [0, 1] → [0, 1] defined by T (x) =3 2 x, 0 ≤ x < 1 2 , 3 2 (1 − x), 1 2 ≤ x ≤ 1.290
Consider the tent map defined by T (x) =2x, 0 ≤ x < 1 2 , 2(1 − x), 1 2 ≤ x ≤ 1.Sketch graphical iterations for the initial conditions (i) x0 = 1 4 , (ii) x0 = 1 6 ,(iii) x0 = 5 7 , and (iv)
Determine an optimal sustainable harvesting policy for the system given in Exercise 9 if the youngest age class is left untouched.
In a fishery, a certain species of fish can be divided into three age groups, each one year long. The Leslie matrix for the female portion of the population is given by L =⎛⎝0 3 36 13 0 0 0 1 2
Assuming the same model for the insects as in Exercise 7, determine the long-term distribution if an insecticide is applied every six months which kills 10% of the youngest age class, 40% of the
Acertain species of insect can be divided into three age classes: 0–6 months, 6–12 months, and 12–18 months. A Leslie matrix for the female population is given by L =⎛⎝0 4 10 0.4 0 0 0 0.2
Linear Discrete Dynamical Systems(a) p(λ) is strictly decreasing;(b) p(λ) has a vertical asymptote at λ = 0;(c) p(λ) → 0 as λ→∞.Prove that a general Leslie matrix has a unique positive
Given that L =⎛⎜⎜⎜⎜⎜⎝b1 b2 b3 · · · bn−1 bn c1 0 0 · · · 0 0 0 c2 0 · · · 0 0...... .... . .......0 0 0 · · · cn−1 0⎞⎟⎟⎟⎟⎟⎠, where bi ≥ 0, 0 < ci ≤ 1,
Consider a human population that is divided into five age classes: those aged 0–15 years, those aged 15–30 years, those aged 30–45 years, those aged 45–60 years, and those aged 60–75 years.
Consider the following Leslie matrix used to model the female portion of a species:L =⎛⎝0 0 6 12 0 0 0 1 3 0⎞⎠.Determine the eigenvalues and eigenvectors of L. Show that there is no dominant
Consider a human population that is divided into three age classes; those aged 0–15 years, those aged 15–30 years, and those aged 30–45 years. The Leslie matrix for the female population is
Solve the following second-order linear difference equations:(a) xn+2 = 5xn+1 − 6xn, n = 0, 1, 2, 3, . . . , if x0 = 1, x1 = 4;(b) xn+2 = xn+1 − 1 4xn, n = 0, 1, 2, 3, . . . , if x0 = 1, x1 =
The difference equation used to model the length of a carpet, say, ln, rolled n times is given by ln+1 = ln + π(4 + 2cn), n = 0, 1, 2, 3, . . . , where c is the thickness of the carpet. Solve this
This is quite a difficult question. Consider the Liénard system(10.10) ˙ x = y − F(x), ˙ y = −x, where F(x) = (a1x + a3x3 + a5x5) is odd. Prove that system (10.10) has at most two limit cycles.
Prove Liénard’s Theorem, that when ∂g = 1 and F(x) is a continuous odd function that has a unique root at x = a and is monotone increasing for x ≥a, (10.5) has a unique limit cycle.
Use Mathematica to investigate the limit cycles, if they exist, of the system˙ x = y − ǫ(a1x + a2x2 +· · ·+aMxM), ˙ y = −x, as the parameter ǫ varies from zero to infinity. When ǫ is
Use the Mathematica notebooks in Chapter 3 to investigate the system˙ x = y − (a1x + a2x2 + a3x3), ˙ y = −x for varying values of the parameters a1, a2, and a3.
Edit the Mathematica notebooks in Chapters 3 and 4 to compare the limit cycles for Liénard systems in the phase plane and in the Liénard plane. Plot the periodic orbits in the xt plane.
Draw a global phase portrait for the Liénard system˙ x = y − x3 − x, ˙ y = −y.
Determine a global phase portrait for the quadratic system given by˙ x = x2 + y2 − 1, ˙ y = 5xy − 5.
Draw a global phase portrait for the system˙ x = −3x + 4y, ˙ y = −2x + 3y and give the equations defining the flow near critical points at infinity.
Draw a global phase portrait for the linear system˙ x = y, ˙ x = −4x − 5y including the flow at infinity.
Using the Mathematica package, investigate the system˙ x = y, ˙ y = x − x3 + ǫ(λy + x2y)when ǫ = 0.1 for values of λ from −1 to −0.5. How many limit cycles are there at most?
Consider the Liénard system given by˙ x = y − ǫ(a1x + a2x2 + a3x3), ˙ y = −x.Prove that for sufficiently small ǫ, there is at most one limit cycle that is asymptotic to a circle of radius r
Plot a phase portrait for the system˙ x = y, ˙ y = x + x2.Determine an equation for the curve on which the homoclinic loop lies.
Consider the generalized mixed Rayleigh–Liénard oscillator equations given by˙ x = y, ˙ y = −x − a1y − b30x3 − b21x2y − b41x4y − b03y3.Prove that at most three small-amplitude limit
Write a program to compute the first five Lyapunov quantities for the Liénard system˙ x = y−(a1x+a2x2+· · ·+a7x7), ˙ y = −(x+b2x2+b3x3+· · ·+b6x6).Prove thatHˆ (4, 2) = 2,Hˆ (7, 2) =
Consider the system˙ x = y − (a1x + a3x3 +· · ·+a2n+1x2n+1), ˙ y = −x.Prove by induction that at most n small-amplitude limit cycles can be bifurcated form the origin.
Write a program to compute the first seven Lyapunov quantities of the Liénard system(9.11) ˙ x = y − (a1x + a2x2 +· · ·+a13x13), ˙ y = −x.Prove that at most six small-amplitude limit cycles
Use Mathematica to compute a Gröbner basis for the set of polynomials{y2 − x3 + x, y3 − x2}under lexicographical, degree lexicographical, and degree reverse lexicographical ordering,
Fix a lexicographical order x ≻ y ≻ z. Divide the multivariate polynomial p = x3+y3+z3 by the ordered list of polynomials {x+3y, xy2−x, y−z}.Repeat the division with the divisors listed as
Prove that the origin of the system˙ x = y − F(G(x)), ˙ y = −G′(x)2 H(G(x))is a center using the transformation u2 = G(x) and the classical symmetry argument.
(a) Consider system (8.10) with k = 0.1 and ω = 1.25. Plot a bifurcation diagram for 0 ≤ Ŵ ≤ 0.12 and show that there is a clockwise hysteresis loop at approximately 0.04 < Ŵ < 0.08. Note that
Adamped driven pendulum may be modeled using the nonautonomous system of differential equations defined by(8.12)d2θdt2 + k dθdt +g lsin(θ) = Ŵ cos(ωt ), where k is a measure of the frictional
Plot the chaotic solution of the periodically driven Fitzhugh–Nagumo system(see Section 4.1)˙u = 10 u − v −u3 3 + I (t), ˙v = u − 0.8v + 0.7, where I (t) is a periodic step function of
Plot three-dimensional and two-dimensional Poincaré sections of the Toda Hamiltonian given by H =p2 12 +p2 22 +e2q2+2√3q1 24 +e2q2−2√3q1 24 +e−4q2 24 −1 8, for several different energy
Consider the two-degrees-of-freedom Hamiltonian given by H(p, q) =ω1 2(p2 1 + q2 1 ) +ω2 2(p2 2 + q2 2 ).Plot three-dimensional and two-dimensional Poincaré sections when(a) ω1 = 3 and ω2 = 7
Use the characteristic multiplier to determine the stability of the limit cycle in Example 4.
Solve the following differential equations:˙r = r(1 − r2), ˙ θ = 1.By considering the line segment = {(x, y) ∈ ℜ2 : 0 ≤ x ≤∞}, find the Poincaré map for this system.
Use the characteristic multiplier to determine the stability of the limit cycle in Example 2.
Obtain a Poincaré map for the system˙ x = μx + y − xx2 + y2, ˙ y = −x + μy − yx2 + y2 on the Poincaré section = {(x, y) ∈ ℜ2 : 0 ≤ x < ∞, y = 0}.
Consider the system˙ x = −y − 0.1xx2 + y2, ˙ y = x − 0.1yx2 + y2.By considering the line segment = {(x, y) ∈ ℜ2 : 0 ≤ x ≤ 4, y = 0}, list the first 10 returns on given that a
Consider system (7.12) given in the text to model the periodic behavior of the Belousov–Zhabotinski reaction. By considering the isoclines and gradients of the vector fields, explain what happens
Plot some time series data for the Lorenz system (7.7) when σ = 10, b = 8 3 , and 166 ≤ r ≤ 167. When r = 166.2, the solution shows intermittent behavior, there are occasional chaotic bursts in
Assume that a given population consists of susceptibles (S), exposed (E), infectives (I), and recovered/immune (R) individuals. Suppose that S +E+I + R = 1 for all time. A seasonally driven epidemic
A three-dimensional Lotka–Volterra model is given by˙ x = x(1−2x+y−5z), ˙ y = y(1−5x−2y−z), ˙z = z(1+x−3y−2z).Prove that there is a critical point in the first quadrant at P( 1 14
The chemical rate equations for the Chapman cycle modeling the production of ozone are O2 + hv → O + O, Rate = k1, O2 + O +M → O3 +M, Rate = k2, O3 + hv → O2 + O, Rate = k3, O + O3 → O2 + O2,
(a) Prove that the origin of the system˙ x = −x − y2 + xz − x3,˙ y = −y + z2 + xy − y3,˙z = −z + x2 + yz − z3 is globally asymptotically stable.(b) Determine the domain of stability
Consider the system˙ x = −x + (λ − x)y, ˙ y = x − (λ − x)y − y + 2z, ˙z =y 2 − z, where λ ≥ 0 is a constant. Show that the first quadrant is positively invariant and that the
Find and classify the critical points of the system˙ x = x − y, ˙ y = y + y2, ˙z = x − z.
Classify the critical point at the origin for the system˙ x = x + 2z, ˙ y = y − 3z, ˙z = 2y + z.7.6. Exercises 167
Find the eigenvalues and eigenvectors of the matrix A =⎛⎝1 0 −4 0 5 4−4 4 3⎞⎠.Hence show that the system ˙x = Ax can be transformed into ˙u = J u, where J =⎛⎝3 0 0 0 −3 0 0 0
Thus far, the analysis has been restricted to bifurcations involving only one parameter, and these are known as codimension-1 bifurcations. This example illustrates what can happen when two
Show that the one-parameter system˙ x = y + μx − xy2, ˙ y = μy − x − y3 undergoes a Hopf bifurcation at μ0 = 0. Plot phase portraits and sketch a bifurcation diagram.
Plot bifurcation diagrams for the planar systems(a) ˙r = r(μ − 0.2r6 + r4 − r2), ˙ θ = −1,(b) ˙r = r((r − 1)2 − μr), ˙ θ = 1.Give a possible explanation as to why the type of
Show that the normal form of a nondegenerate Hopf singularity is given by˙u˙v=0 −ββ 0 uv+au(u2 + v2) − bv(u2 + v2)av(u2 + v2) + bu(u2 + v2)+O(|u|5), whereβ > 0 and a = 0.
Determine the nonlinear transformation which eliminates terms of degree three from the planar system˙ x = λ1x + a30x3 + a21x2y + a12xy2 + a03y3,˙ y = λ2y + b30x3 + b21x2y + b12xy2 + b03y3, where
Consider the following one-parameter systems of differential equations in polar form:(a) ˙r = μr(r + μ)2, ˙ θ = 1;(b) ˙r = r(μ − r)(μ − 2r), ˙ θ = −1;(c) ˙r = r(μ2 − r2), ˙ θ =
Consider the following one-parameter systems of differential equations:(a) ˙ x = x, ˙ y = μ − y4;(b) ˙ x = x2 − xμ2, ˙ y = −y;(c) ˙ x = −x4 + 5μx2 − 4μ2, ˙ y = −y.Find the
Acertain species of fish in a large lake is harvested. The differential equation used to model the population, x(t) in hundreds of thousands, is given by dx dt = x1 −x 5−hx 0.2 + x.Determine and

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