Questions and Answers of Fundamentals Of Vehicle Dynamics And Modelling

Use the Lyapunov function V (x, y) = x4 + 2y2 − 10 to investigate the invariant sets of the system˙ x = y − x(x4 + 2y2 − 10), ˙ y = −x3 − 3y5(x4 + 2y2 − 10).
Plot a phase portrait for the system in Exercise 8.
Determine the basin of attraction of the origin for the system˙ x = x(x2 + y2 − 4) − y, ˙ y = x + y(x2 + y2 − 4)using the Lyapunov function V (x, y) = x2 + y2.
Determine the values of a for which V (x, y) = x2 + ay2 is a Lyapunov function for the system˙ x = −x + y − x2 − y2 + xy2, ˙ y = −y + xy − y2 − x2y.
Prove that the origin is a unique critical point of the system˙ x = −1 2y(1+x)+x(1−4x2−y2), ˙ y = 2x(1+x)+y(1−4x2−y2).Determine the stability of the origin using the Lyapunov function V
Investigate the stability of the critical points at the origin for the following systems:(a) ˙ x = −y − x3, ˙ y = x − y3, using the Lyapunov function V (x, y) =x2 + y2;(b) ˙ x = x(x − α),
Plot a phase portrait of the system˙ x = y(y2 − 1), ˙ y = x(1 − x2).
Plot a phase portrait for the damped pendulum equation¨ θ + 0.15 ˙ θ + sin θ = 0 and describe what happens physically.
Given the Hamiltonian function H(x, y) = y2 2 + x2 2 − x4 4 , sketch a phase portrait for the Hamiltonian system.
Find the Hamiltonian of the system˙ x = y, ˙ y = x − x3 and sketch a phase portrait.
Using the method of multiple scales, show that the one-term uniform valid expansion of the ODE d2x dt2 + x = −ǫdx dt, with initial conditions x(0) =b, ˙ x(0) = 0, is x(t, ǫ) ∼ xMS = be−ǫt 2
Use the Lindstedt–Poincaré technique to obtain a one-term uniform expansion for the ODE d2x dt2 + x = ǫx1 −dx dt 2, with initial conditions x(0) = a and ˙ x(0) = 0.
Prove that neither of the following systems have limit cycles using the given multipliers:(a) ˙ x = x(4 + 5x + 2y), ˙ y = y(−2 + 7x + 3y), ψ = 1 xy2 .(b) ˙ x = x(β − δx − γy), ˙ y = y(b
Prove that none of the following systems have limit cycles:(a) ˙ x = y, ˙ y = −x − (1 + x2 + x4)y;(b) ˙ x = x − x2 + 2y2, ˙ y = y(x + 1);(c) ˙ x = y2 − 2x, ˙ y = 3 − 4y − 2x2y;(d)
Plot phase portraits for the Liénard system˙ x = y − μ(−x + x3), ˙ y = −x, when (a) μ = 0.01, and (b) μ = 10.
For which parameter values does the Holling–Tanner model˙ x = xβ1 −x k−rxy(a + ax), ˙ y = by 1 −Ny xhave a limit cycle?
Prove that the system˙ x = y + x(α − x2 − y2), ˙ y = −x + y(1 − x2 − y2), where 0 < α < 1, has a limit cycle and determine its stability.
Prove that the system˙ x = x − y − x3, ˙ y = x + y − y3 has a unique limit cycle.
By considering the flow across the square with coordinates (1, 1), (1,−1),(−1,−1), (−1, 1), centered at the origin, prove that the system˙ x = −y + x cos(πx), ˙ y = x − y3 has a stable
Prove that the system˙ x = y + x 12 − x2 − y2, ˙ y = −x + y(1 − x2 − y2)has a stable limit cycle. Plot the limit cycle.
The following three differential equations are used to model a combined predator–prey and competing species system:˙ x = x(a10 − a11x + a12y − a13z),˙ y = y(a20 − a21x − a22y −
Suppose that there are three species of insect X, Y , and Z, say. Give rough sketches to illustrate the possible ways in which these species can interact with one another. You should include the
Use Mathematica to plot a trajectory for the predator–prey system˙ x = x(x − x2 − y), ˙ y = y(x − 0.48)using the initial condition (0.6, 0.1). What can you deduce about the longterm
A predator–prey model is given by˙ x = x(x − x2 − y), ˙ y = y(x − 0.6).Sketch a phase portrait and interpret the results in physical terms.
A predator–prey system may be modeled using the differential equations˙ x = x(1 − y − ǫx), ˙ y = y(−1 + x − ǫy), where x(t) is the population of prey and y(t) is the predator population
(a) Sketch a phase portrait for the system˙ x = x(4 − y − x), ˙ y = y(3x − 1 − y), x ≥ 0, y≥ 0, given that the critical points occur at O = (0, 0), A = (4, 0), and B = (5/4, 11/4).(b)
The differential equations used to model a competing species are given by˙ x = x(2 − x − y), ˙ y = y(μ − y − μ2x), where μ is a constant. Describe the qualitative behavior of this system
Plot a phase portrait for the following system and describe what happens to the population for different initial conditions:˙ x = 2x − x2 − xy, ˙ y = −y − y2 + xy.
Plot a phase plane diagram for the following predator–prey system and interpret the solutions in terms of species behavior:˙ x = 2x − xy, ˙ y = −3y + xy.
Plot a phase portrait for the following competing species model:˙ x = 2x − x2 − xy, ˙ y = 3y − y2 − 2xy and describe what happens in terms of species behavior.
Given that d3ηdτ3 = −ηd2ηdτ2 and x =η dηdτd2ηdτ2, y=dηdτ2η d2ηdτ2, and t = logdηdτ, prove that˙ x = x(1 + x + y), ˙ y = y(2 + x − y).Plot a phase portrait in the xy
A very simple model for the economy is given by I˙ = I − KS, S˙ = I − CS − G0, where I represents income, S is the rate of spending, G0 denotes constant government spending, and C and K are
The power, say, P, generated by a water wheel of velocity V can be modeled by the system P˙ = −αP + PV, V˙ = 1 − βV − P2, where α and β are both positive. Describe the qualitative
An age-dependent population can be modeled by the differential equations p˙ = β + p(a − bp), β˙ = β(c + (a − bp)), where p is the population, β is the birth rate, anda, b, and c are all
A nonlinear capacitor–resistor electrical circuit can be modeled using the differential equations˙ x = y, ˙ y = −x + x3 − (a0 + x)y, where a0 is a nonzero constant and x(t) represents the
Construct a nonlinear system that has four critical points: two saddle points, one stable focus, and one unstable focus.
Plot phase portraits for the following systems:(a) ˙ x = y, ˙ y = x − y + x3;(b) ˙ x = −2x − y + 2, ˙ y = xy;(c) ˙ x = x2 − y2, ˙ y = xy − 1;(d) ˙ x = 2 − x − y2, ˙ y = −y(x2
Avery simple mechanical oscillator can be modeled using the second-order differential equation d2x dt2 + μdx dt + 25x = 0, where x measures displacement from equilibrium.(a) Rewrite this equation as
Sketch phase portraits for the following linear systems:(a) ˙ x = 0, ˙ y = x + 2y;(b) ˙ x = x + 2y, ˙ y = 0;(c) ˙ x = 3x + 4y, ˙ y = 4x − 3y;(d) ˙ x = 3x + y, ˙ y = −x + 3y;(e) ˙ x = y,
(a) Find the eigenvalues and eigenvectors of the matrix B =−7 6 2 −6.Sketch a phase portrait for the system ˙x = Bx and its corresponding canonical form.(b) Carry out the same procedures as in
(a) Determine the maximal interval of existence for each of the following initial value problems:(i) ˙ x = x4, x(0) = 1;(ii) ˙ x = x2−1 2 , x(0) = 2;(iii) ˙ x = x(x − 2), x(0) = 3.(b) For what
In an epidemic the rate at which healthy people become infected is a times their number, the rates of recovery and death are, respectively, b and c times the number of infected people. If initially
Two tanks A and B, each of volume V , are filled with water at time t = 0.For t > 0, volume v of solution containing mass m of solute flows into tank A per second; mixture flows from tank A to tank B
Achemical substanceAchanges into substanceB at a rate α times the amount of A present. Substance B changes into C at a rate β times the amount of B present. If initially only substance A is present
The differential equation used to model the concentration of glucose in the blood, say, g(t), when it is being fed intravenously into the body, is given by dg dt + kg =G 100V, where k is a constant,
Aforensic scientist is called to the scene of a murder. The temperature of the corpse is found to be 75◦F and one hour later the temperature has dropped to 70◦F. If the temperature of the room in
(a) Consider a series resistor–inductor circuit with L = 2H, R = 10, and an applied EMF of E = 100 sin(t). Use an integrating factor to solve the differential equation, and find the current in the
Write down the chemical reaction rate equations for the reversible reaction equations(a) A + B + C ⇋ D,(b) A + A + A ⇋ A3, given that the forward rate constant is kf and the reverse rate constant
Fossils are often dated using the differential equation dA dt = −αA, where A is the amount of radioactive substance remaining, α is a constant, and t is measured in years. Assuming that α =
Sketch some solution curves for the following differential equations:(a) dy dx = −y x ;(b) dy dx = 2y x ;(c) dy dx = y 2x ;(d) dy dx = y2 x ;(e) dy dx = − xy x2+y2 ;(f) dy dx = y x2 .
(a) Asix-neuron discrete Hopfield network is required to store the following fundamental memories:x1 = (1, 1, 1, 1, 1, 1)T , x2 = (1,−1, 1,−1,−1, 1)T , x3 = (1,−1,−1, 1,−1, 1)T .(i)
(a) Given the complex mapping En+1 = A + BEnei|En|2 , determine the number and approximate location of fixed points of period one when A = 3.2 and B = 0.3.[10](b) Edit the given program for producing
(a) Edit the given program for plotting a bifurcation diagram for the logistic map (see Chapter 12) to plot a bifurcation diagram for the tent map.[10](b) Write a program to plot a Julia set J(0,
(a) Find and classify the fixed points of period one for the Hénon map xn+1 = 1.5 + 0.2yn − x2 n, yn+1 = xn.Find the approximate location of fixed points of period two if they exist. Plot a
(a) Given that f (x) = 3.5x(1 − x),(i) plot the graphs of f (x), f 2(x), f 3(x), and f 4(x);(ii) approximate the fixed points of periods one, two, three, and four, if they exist;(iii) determine the
(a) For the system dx dt = μx + x3, dy dt = −y sketch phase portraits forμ < 0, μ = 0, andμ > 0. Plot a bifurcation diagram.[10](b) Plot a phase portrait and Poincaré section for the forced
(a) Two solutes X and Y are mixed in a beaker. Their respective concentrations x(t) and y(t) satisfy the following differential equations:dx dt = x − xy − μx2, dy dt = −y + xy − μy2.Find
(a) The radioactive decay of polonium 218 to bismuth 214 is given by 218Po → 214Pb → 214Bi, where the first reaction rate is k1 = 0.5 s−1, and the second reaction rate is k2 = 0.06 s−1.(i)
(a) A four-neuron discrete Hopfield network is required to store the following fundamental memories:x1 = (1, 1, 1, 1)T , x2 = (1,−1, 1,−1)T , x3 = (1,−1,−1, 1)T .(i) Compute the synaptic
(a) Find and classify the fixed points of period one of the Hénon map defined by xn+1 = 1 −9 5x2 n + yn, yn+1 =1 5xn.[8]424 18. Examination-Type Questions(b) Consider the complex iterative
(a) Starting with an equilateral triangle (each side of length 1 unit) construct the inverted Koch snowflake up to stage 2 on graph paper.At each stage, each segment is 1 3 the length of the previous
(a) Let T be the function T : [0, 1] → [0, 1] defined by T (x) =7 4 x, 0 ≤ x < 1 2 , 7 4 (1 − x), 1 2 ≤ x ≤ 1.Determine the fixed points of periods one, two, and three.[12](b) Determine
(a) Consider the two-dimensional system dr dt = r(μ − r)(μ − r2), dθdt = −1.Show how the phase portrait changes as the parameter μ varies and draw a bifurcation diagram.[10](b) Prove that
(a) Find the eigenvalues of the following system and sketch a phase portrait in three-dimensional space:dx dt = −2x − z, dy dt = −y, dz dt = x − 2z.[12](b) Show that the origin of the
(a) Prove that the origin of the system dx dt = −x 2 + 2x2y, dy dt = x − y − x3 is asymptotically stable using the Lyapunov function V = x2 + 2y2.[6](b) Solve the differential equations dr dt =
(a) Sketch a phase portrait for the following system showing all isoclines:dx dt = 3x + 2y, dy dt = x − 2y.[6](b) Show that the system dx dt = xy − x2y + y3, dy dt = y2 + x3 − xy2 can be
Use the OGY method to control chaos in the minimal chaotic neuromodule.
Consider the neuromodule defined by the equations xn+1 = 2 + 3.5 tanh(x) − 4 tanh(0.3y), yn+1 = 3 + 5 tanh(x).Iterate the system and show that it is quasiperiodic.
A simple model of a neuron with self-interaction is described by Pasemann[12]. The difference equation is given by an+1 = γan + θ + wσ(an), 0 ≤ γ < 1, where an is the activation level of the
Add suitable characters “3’’ and “5’’ to the fundamental memories shown in Figure 17.12. You may need to increase the grids to 10 × 10 and work with 100 neurons.
Consider the discrete Hopfield model investigated in Example 5. Test the vector x7 = (−1,−1, 1, 1, 1)T , update in the following orders, and determine to which vector the algorithm converges:(a)
(a) ProveTheorem 3 regardingLyapunov functions of continuous Hopfield models.416 17. Neural Networks(b) Consider the recurrent Hopfield network modeled using the differential equations˙ x = −x + 7
Use another data set of your choice from the URL http://www.ics.uci.edu/∼mlearn/MLRepository.html using an edited version of the programs listed in Section 17.5 to carry out your analysis.
By editing the programs listed in Section 17.5:(a) Investigate what happens to the mean squared error for varying eta values of your choice.(b) Investigate what happens to the mean squared error as
Prove Theorem 2, showing that when the activation function is nonlinear, say, yk = φ(vk), the generalized delta rule can be formulated as wkj (n + 1) = wkj (n) − ηgkj , where gkj = (yk −
For the following activation functions, show that(a) if φ(v) = 1/(1 + e−av), then φ′(v) = aφ(v)(1 − φ(v));(b) if φ(v) = a tanh(bv), then φ′(v) = b a (a2 − φ2(v));(c) if φ(v) = 1 2a
Use the methods described in Section 16.4 to demonstrate synchronization of chaos in Chua’s circuit.
Try the same procedure of control to period one for the Ikeda map as in Exercise 8 but with the parameters A0 = 7 and B = 0.15. Investigate the size of the control region around one of the fixed
Use the OGY method (see Section 16.3) with the parameter A to control the chaos to a point of period one in the Ikeda map En+1 = A + BEnei|En|2 when A0 = 2.7 and B = 0.15. Display the control on a
Plot the chaotic attractor for the Ikeda map En+1 = A + BEnei|En|2 when Recommended Reading 383(i) A = 4 and B = 0.15;(ii) A = 7 and B = 0.15.How many points are there of period one in each case?
Consider the Ikeda map, introduced in Chapter 12, given by En+1 = A + BEnei|En|2.Suppose that En = xn + iyn. Rewrite the Ikeda map as a two-dimensional map in xn and yn. Plot the chaotic attractor
Use the OGY algorithm given in Section 16.3 to stabilize a point of period one in the Hénon map xn+1 = a + byn − x2 n, yn+1 = xn when a = 1.4 and b = 0.4. Display the control using a time series
Apply the method of chaos control by periodic proportional pulses (see Section 16.2) to the two-dimensional Hénon map xn+1 = a + byn − x2 n, yn+1 = xn, where a = 1.4 and b = 0.4. (In this case,
Find the points of periods one and two for the Hénon map given by xn+1 = a + byn − x2 n, yn+1 = xn when a = 1.4 and b = 0.4, and determine their type.
Apply the method of chaos control by periodic proportional pulses (see Section 16.2) to the logistic map xn+1 = μxn(1 − xn)when μ = 3.9. Sketch the graphs Ci(x), i = 1 to 4. Plot time series data
Show that the map defined by xn+1 = 1 + yn − ax2 n, yn+1 = bxn can be written as un+1 = a + bvn − u2 n, vn+1 = un using a suitable transformation.382 16. Chaos Control and Synchronization
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

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