double pendulum 309
15.1 Anharmonic potential V{x) = —{a/2)x^ + (6/4)x^, for a = —A
(left figure) and a = 4 (right figure). In both cases b = 0.05 312 15.2 Solution of the Duffing equation in the interval [0,30], for
a = —4 and b = 0.05, and the initial conditions x(0) = —10
and x'{0) =0 313 15.3 Solution of the Duffing equation in the interval [0,30], for
a = 4: and b = 0.05, and the initial conditions x(0) = 0 and 17.1 Equipotentials, in the plane z = 0.01, of a unit electric charge
located at the origin 329 17.2 Electric field created by a unit electric charge located at the
origin 330 17.3 Electric field created by a unit dipole, represented by a bigger
arrow, located at the origin 331 17.4 Equipotentials and electric field lines created by three charges
respectively equal to +2 localized at the origin and —1 localized
on the Ox-axis at a distance —1/2 and 1/2 from the origin 334 17.5 Equipotentials and electric field lines created by four charges
respectively equal to —1, + 1 , —1 and + 1 localized at the
vertices of a unit square centered at the origin 335 17.6 Equipotentials and electric field lines created by three charges
respectively equal to +2 localized at the origin and two negative
unit charges localized at (—1/2, —1/2,0) and (1/2, —1/2,0) 336 17.7 Electric field created by a uniformly charged sphere as a
function of the distance r from the sphere center 337 17.8 Electric potential created by a uniformly charged sphere as a
function of the distance r from the sphere center. 339
xxvi List of Figures
19.1 Graphs of Li and L2, the first two steps in the construction of
the Lebesgue function L 353 19.2 Graph of L3 the third step in the construction of the Lebesgue
function L 354 19.3 First stage in the construction of the Sierpinski triangle 355
19.4 Second stage in the construction of the Sierpinski triangle 356 19.5 Fifth stage in the construction of the Sierpinski triangle 357 19.6 First stage in the construction of the Sierpinski square 359 19.7 Fifth stage in the construction of the Sierpinski square 359 19.8 First stage of the construction of the von Koch curve 360 19.9 Second stage of the construction of the von Koch curve 362 19.10 Second stage of the construction of the von Koch curve using
lineSequence instead of the listable version of the function
nextProf i l e 363 19.11 Fourth stage of the construction of the von Koch curve 364
19.12 Fifth stage of the construction of the von Koch curve 364 19.13 Same as above but starting from a different set of points 365 19.14 Fifth stage of the construction of the von Koch triangle 365 19.15 Fourth stage of the construction of the von Koch square 367 20.1 Sequence of points generated by the chaos game starting from
an initial point (labeled 1) inside an equilateral triangle 371 20.2 Sequence of points generated by the chaos game starting from
an initial point (labeled 1) outside the triangle 371 20.3 The sequence of a large number of points generated by the
chaos game seems to converge to a Sierpinski triangle 372 20.4 Sequence of a large number of points generated by the chaos
game of Example 1 374 20.5 Sequence of a large number of points generated by the chaos
game of Example 2 375 20.6 Sequence of a large number of points generated by the chaos
game of Example 3 376 20.7 Bamsley's fern 378
20.8 Barnsley^s fern with the fixed points of the affine
transformations fi, f2, fs, and f^ 378 20.9 Action of the four affine transformations on the initial shape.
Upper left: / i generates the lower part of the stem. Upper right: /2 generates the upper part of the stem, all triangles converging to the fixed point 2 o/ /2. Lower left: starting from the image of the initial shape by fs, and repeatedly applying /2 generates the left branches. Lower right: starting from the image of the initial shape by f4, and repeatedly applying /2
generates the right branches 380 20.10 Bamsley's fern with probabilities pi = 0.03, p2 = 0.75,
p^=p^ = 0.11 381 20.11 Leaflike fractal generated using Bamsley^s collage theorem 383
21.1 Julia set of the function z^^ z^ — 0.5 386 21.2 Julia set of the function z\-^ z^ — 0.75 + 0.5i 387 21.3 Julia set above: zooming in [0.9,1.6] x [—0.7, —0.1] 388 21.4 Julia set above: zooming in [1.26,1.28] x [-0.2, -0.1] 388 21.5 Julia set above: zooming in [1.24,1.27] x [-0.13, -0.1] 389
21.6 Julia set of the function z \-^ z^ — 0.5 390 21.7 Julia set of the function z \-^ z^ — 0.75 -h 0.5i 390 21.8 Julia set above: zooming in [—0.9,0.1] x [0.1,1.3] 391 21.9 Julia set above: zooming in [-0.57, -0.38] x [0.9,1.25] 391
21.10 Julia set of the function z y-^ z^ — 0.5 392 21.11 Julia set above: zooming in [-0.1,0.1] x [1.02,1.22] 392
21.12 Mandelbrot set of the function z ^-^ z'^ -\- c 393 21.13 Mandelbrot set: zooming in [-1.0, -0.4] x [-0.3,0.3] 394
21.14 Mandelbrot set: zooming in [-0.85, -0.65] x [-0.2,0] 395 21.15 Mandelbrot set: zooming in [-0.77, -0.72] x [-0.2, -0.15] 395 21.16 Mandelbrot set: zooming in [-0.748,-0.74] x [-0.186,-0.178]. 396
21.17 Mandelbrot set for the function z \-^ z^ -\- c 397 21.18 Mandelbrot set for the function z y-^ z^ -\- c 398 22.1 Elliptical orbits. The big dot represents the sun 403
xxviii List of Figures
22.2 Hyperbolic orbit. The big dot represents the sun 403 23.1 Fourth stage of the construction of the von Koch curve 410 23.2 Fourth stage of the construction of the von Koch triangle 411 23.3 Sixth stage of the construction of the Hilbert curve 413 23.4 First stage of the construction of the Peano curve 415 23.5 Third stage of the construction of the Peano curve 416 24.1 Logistic map cobweb for r = 2.6, UQ = 0.9, and a number of
iterations equal to 15 420 24.2 Sixteen iterations of the logistic map for r — 2.3 (fixed point),
r = 3.23 (period 2), r = 3.49 (period 4), CL'f^d r = 3.554 (period
8) 425 24.3 Bifurcation diagram of the logistic map {n,r) i—> r n ( l — n).
The parameter r, plotted on the horizontal axis, varies from 2.5 to 4, and the reduced population n, plotted on the vertical
axis, varies between 0 and 1 426 24.4 Logistic map cobweb for r = 4, UQ = y/S — 1, and a number of
iterations equal to 300. The initial point is defined with 200
significant digits 427 24.5 Approximate cumulative distribution function for the logistic
map n i-^ 4n(l — n) 428 24.6 One hundred iterates of the logistic map n ^-^ 4n(l — n) starting
from no = sin^{2Ti/7), defined with $MachinePrecision,
showing the instability of the period-3 point 432 24.7 One hundred iterates of the logistic map n i-^ 4n(l — n),
starting from UQ = sm^(27r/7) defined with 70 significant digits. 433 24.8 Invariant probability density of the logistic map n i-> 4n(l — n ) . . 436 24.9 Invariant cumulative distribution function of the logistic map
n ^ 4n(l - n) 436 24.10 Comparing the exact invariant cumulative distribution
function (in blue) with the approximate one (in red) obtained
above. The two curves cannot be distinguished 437
25.1 Projection on the xOy-plane of a numerical solution of the
Lorenz equations for t G [0,40] and (XQ, yo^ 2:0) = (0,0,1) 440
25.2 Projection on the yOz-plane of a numerical solution of the
Lorenz equations for t G [0,40] and (XQ, yo, ^o) = (0? 0? 1) 441 25.3 Projection on the xOz-plane of a numerical solution of the
Lorenz equations for t G [0,40] and (XQ, yo, ZQ) = (0,0,1) 441 25.4 Projection on the yOz-plane showing the trajectory slowly
moving away from the unstable fixed points 443
26.1 The Morse potential (in red) and its harmonic part (in blue). . . 446
29.1 Construction of the quadratrix) 472 29.2 Construction of a segment of length I/TT 473
29.3 Construction of a segment of length ^TT 474 30.1 Hermite polynomials Hi, H2, Hs, and H4 477 30.2 Normed wave functions ipo and ipi 478 30.3 Normed wave functions 1^2 o,nd ips 478 30.4 Normed wave functions -^4 and ijj^ 478 31.1 Graphs of the functions y 1—> \f\ — y^jy and y i-^ t a n y in the
interval [0,4] 483 31.2 Graphs of the functions y H-> yj\ — y^ jy and y 1-^ — cot^/ in
the interval [0,4] 484 31.3 Square potential well and energy levels 485
31.4 Eigenfunction associated with the energy level
El = -0.901976 Vo 486 31.5 Eigenfunction associated to the energy level E^ = —0.617279 VQ- 487
31.6 Eigenfunction associated with the energy level
Es = -0.192111 Vo 488 32.1 Free-fall diverts velocity as a function of time 490
32.2 Diver's velocity as a function of time when parachute opening
is delayed 491 32.3 Rapid change of the diver's velocity when the parachute takes
less than one second to fully open 491
XXX List of Figures
32.4 Diverts acceleration when the parachute is opened in a very
short time 492 32.5 Diver's velocity when the parachute takes three seconds to fully
open 493 32.6 More detailed plot of the diver's velocity when the parachute
takes three seconds to fully open 494 32.7 Diver's acceleration when the parachute takes three seconds to
fully open 495
33.1 The evolute of a cycloid is a cycloid 498
33.2 Huygens pendulum 499 34.1 Trajectory in the {x\^X2)-phase space of the van der Pol
oscillator for A = - 0 . 5 and t G [0,30] 506 34.2 Trajectory in the {xi^X2)-phase space of the van der Pol
oscillator for A = 0.5 and t G [0,50] 507
35.1 Dimensionless van der Waals isotherms 513
35.2 Maxwell construction 517 36.1 Initial pedestrian configuration. Type 1 pedestrians (blue
squares) move to the right, and type 2 (red squares) move to
the left 527 36.2 Final pedestrian configuration. Type 1 pedestrians (blue
squares) move to the right, and type 2 (red squares) move to
the left 527