Skiadas C.H., Skiadas C. Chaotic Modelling and Simulation. Analysis of Chaotic Models, Attractors and Forms

Подождите немного. Документ загружается.

14 Chaotic Modelling and Simulation

(a) Fifth-order symmetry (b) Multiple-order symmetries

FIGURE 1.13: Symmetric forms

In Chapter 9, several symmetric and non-symmetric forms are presented and anal-

ysed. The following ﬁgures illustrate a symmetric chaotic form before the separation

step (Figures 1.14(a) and 1.14(b)), in the intermediate stage (Figure 1.14(c)), and af-

ter the separation of the chaotic form into two symmetric forms (Figure 1.14(d)).

The parameters are b = 1, c = 2.85 and d = 16, and the parameter a takes the values

a = 18, 35, 45 and 55 respectively. The rotation iterative procedure is used, with

rotation angle θ

= c +

Another category of patterns includes forms which we call “ﬂowers.” By using

the rotation formula with rotation angle

= c +

and with parameter values b = 0.9, a = c = d =

2π

, k =

and δ = 2, the graph in

Figure 1.15(a) appears. When a =

2π

, c = d =

2π

and δ = 3, Figure 1.15(b) results.

1.4 Chaos and the Universe

1.4.1 Chaos in the solar system

Complicated and even chaotic orbits and paths in the solar system are studied

and simulated in Chapter 11. Objects in the solar system mainly follow circular or

elliptic orbits around a centre of mass. In this case the simulation can follow a purely

geometric approach. To simplify the problem the centre of mass is assumed to be

located in the centre of the large mass. The attracting force is, of course, gravity, and

Introduction 15

(a) One symmetric form (b) Starting separation

FIGURE 1.14: Separation of chaotic forms

16 Chaotic Modelling and Simulation

(a) δ = 2 (b) δ = 3

FIGURE 1.15: Flowers

is presented by an inverse square law of the distance. The non-dimensional equations

of motion for a body with small mass m rotating around a large body with mass M,

which we consider to stay in equilibrium position, are, in Cartesian coordinates:

¨x = −



+ y



3/2

¨y = −



+ y



3/2

(1.16)

The initial conditions are x = 1, y = 0, ˙x = 0 and ˙y = ν. The escape velocity is

ν =

√

2, and the circular orbit is obtained when ν = 1. The circular orbit is illustrated

in Figure 1.16(a). In the same ﬁgure, an elliptic orbit is presented, where the initial

velocity is 1 < ν = 1.2 <

√

2 . A hyperbolic orbit is also presented. The initial

velocity is higher than the escape limit



ν = 1.45 >

√



and the particle ﬂies oﬀ to

inﬁnity. Figure 1.16(b) illustrates a large number of elliptic paths for the same value

of the initial velocity as before. The paths remain in the same plane, but deviate from

the original position, forming the shape presented in the ﬁgure.

The motions of two attracting point masses are relatively simple. The equations

of motion for the point mass m in Cartesian coordinates are:

¨x

= −

− x



− x

)

+ (y

− y

)



3/2

¨y

= −

− y



− x

)

+ (y

− y

)



3/2

(1.17)

Analogous is the case for the point mass m

. The gravity constant is G = 1 and

the total mass of the rotating system is M = m

+ m

. If we give the two masses

initial positions (x

, y

) and (x

, y

), and initial velocities ˙x

, ˙y

, ˙x

and ˙y

, then the

Introduction 17

(a) Circular, Elliptic and Hyperbolic orbits (b) Many elliptic orbits

FIGURE 1.16: Orbits in two-body motion

masses revolve around each other while drifting together, as a pair, through space.

Their centre of mass drifts at a constant velocity. Relative to an observer moving

with the centre of mass, their orbits are ellipses and, in special cases, circles. The

motion of two attracting points in space is presented in Figure 1.17(a). The heavy line

represents the movement of the centre of mass. The masses revolve around the centre

and drift together. The points have masses m

= 0.4, m

= 0.6 and initial velocities

˙x

= −0.15, ˙x

= 0, ˙y

= −0.45 and ˙y

= 0.25. The same motion, viewed relative

to the centre of mass, is illustrated in Figure 1.17(b). Each point-mass follows an

elliptic path.

(a) Attracting bodies in space (b) Relative movement

FIGURE 1.17: Motion in the plane

18 Chaotic Modelling and Simulation

The three-body problem, reduced to the plane, still gives very complicated and

chaotic orbits. To simplify the problem we assume that the third body, a satellite, is

of negligible mass relative to the other two bodies. The other two, the planets, travel

around the centre of the combined mass in circular orbits in the same plane. The

satellite rotates in the same plane. The total mass of the planets is m

+ m

= 1. The

equations of motion for the satellite then are:

¨x = −

(x − x

)

3/2

−

(x − x

)

3/2

¨y = −

(y − y

)

3/2

−

(y − y

)

3/2

(1.18)

where (x, y) is the position of the satellite, r

, r

are the distances of the satellite from

the two planets, m

, m

are the masses of the two planets, and (x

, y

) and (x

, y

) are

the positions of the planets at the same time t.

In the following example the planets have masses m

= 0.9, , m

= 0.1, and the

position and initial velocity of the satellite are (−1.033, 0) and (0, 0.35) respectively.

The complicated and chaotic paths are illustrated in the following ﬁgures. In Fig-

ure 1.18(a), the movement is viewed from the outside of the system, from space. It

shows that the satellite can move between the two revolving planets. Figure 1.18(b)

illustrates the orbit of the satellite viewed in a coordinate system that rotates with the

revolution of the planets (rotating frame), with the X-axis always passing through the

planets according to the rotation formulas:

X = x cos t + y sin t

Y = −x sin t + y cos t

(1.19)

(a) Motion in space (b) Motion in a rotating frame

FIGURE 1.18: Motion in space

Introduction 19

By using diﬀerence equations to model solar systems, it is possible to simulate

chaotic patterns like the rings of Saturn and other planets. The simplest approach is

to use an equation like the logistic, and to assume that the system rotates following

elliptic orbits. Two realizations appear in the next ﬁgures. The parameter of the

logistic model is b = 3.6 for Figure 1.19(a) and b = 3.68 for Figure 1.19(b). The

rotation formula is:

= 2r

cos t

= 2r

sin t

(1.20)

where r

t+1

= br

(1 − r

(a) b = 3.6 (b) b = 3.68

FIGURE 1.19: Ring systems

1.4.2 Chaos in galaxies

The underlying literature on galactic simulations deals mainly with the N-body

problem and related simulations. The approach to the N-body problem follows some

simpliﬁcations. The majority of the models we examined are based on numerical

integration of the equations of motion for N mutually interacting particles with equal

masses, which follow a steady rotational motion.

The acceleration of the i-th parti-

cle is:

j,i

− r



i j

+ r

cut



3/2

See Sellwood (1983, 1989); Sellwood and Wilkinson (1993).

20 Chaotic Modelling and Simulation

The cutoﬀ radius r

cut

greatly simpliﬁes the numerical computation by eliminating

the infrequent, but very large, accelerations at close encounters. The gravity constant

and the mass of each particle are set to 1. When N is large, the main computational

problem is the large number (∝ N(N − 1)) of operations required to determine the

accelerations.

When a small perturbation is applied to the rotating system, familiar galaxy forms

appear after some iterations. The details are presented in Chapter 11. Two character-

istic examples of these forms are the two-armed spirals illustrated in Figures 1.20(a)

and 1.20(b). The simulations start from an elliptic disk with diﬀerent density dis-

tribution. In Figure 1.20(b), the high central density leads to the formulation of a

barred spiral form.

(a) Two-armed spiral



ρ =

, t = 100



(b) Barred spiral



ρ =

, t = 200



FIGURE 1.20: Spiral galaxies

Another approach to producing a spiral pattern formation is based on the elliptical

epicycle that a star follows around its guiding centre. According to the theory of

kinematic spirals, the oval orbits are nested to form a two-armed spiral pattern of

a galaxy.

The simulation results, illustrated in Figure 1.21(a), are based on the

following equation in polar coordinates (r, θ):

r =

1 + a cos



n(5 − 5R

+ φ)



The values of the guiding centre are 0.2 < R

< 1. The parameters are a =

0.08, n = 2 for the two-armed spiral and a = 0.15, n = 1 for the one-armed spiral

(Figure 1.21(b)).

See also Sparke and Gallagher (2000).

Introduction 21

(a) Two-armed spiral (b) One-armed spiral

FIGURE 1.21: Spiral galaxies

Recent discoveries of new galaxies, intergalactic forms, planetary nebulae, black

holes and other objects give inspiration for new simulations of the large variety of

observed forms and shapes. The questions about black matter and how it inﬂuences

the movement and shape of galaxies and other objects change our approach to model

building and the related simulations of galactic forms. In many instances, the galactic

forms arise by using very simple models and simulation tools. The forms presented

below arise from the reﬂection iterative formula (1.20) with parameters a = 1, b = 1,

and rotation angle:

= c −

The reﬂection procedure leads to mirror image symmetric shapes. In some in-

stances a central chaotic bulge is created. Figure 1.22(a) illustrates the beginning of

the mirror image formation, whereas in Figure 1.22(b) the central bulge is already

present. The outer part has the form of an electromagnetic ﬁeld. In both cases the

parameter c is set to 1.6, while the parameter d is 35 for Figure 1.22(a) and 200 for

Figure 1.22(b).

When d = 800, the reﬂection image takes the form presented in Figure 1.22(c).

There is a central-bulge connected with the outer periphery by two symmetric routes.

Figure 1.22(d) also illustrates a case where d = 800, but now a = 0.6. In this case,

only the central bulge remains, while the form in the outer region is just beginning

to take shape.

Rotations, reﬂections and translations with appropriate selection of the equation

of the rotation angle give rise to a plethora of galaxy formations. Figure 1.23(a)

illustrates a simple spiral pattern with one companion, whereas in Figure 1.23(b)

two spiral galaxies with two companions appear. In both cases b = 0.9, c = 2, d = 6,

and a is 0.3 for Figure 1.23(a) and 5 for Figure 1.23(b). The rotation angle in both

22 Chaotic Modelling and Simulation

(a) d = 35 (b) d = 200

FIGURE 1.22: Reﬂection forms

Introduction 23

case is given by:

= c +

(a) a = 0.3 (b) a = 5

FIGURE 1.23: Spiral Patterns

1.4.3 Galactic-type potentials and the H´enon-Heiles system

Contopoulos (1958, 1960, 1965) explored the box-like paths of a body in a galactic

potential, and found that the orbits in the meridian plane of an axisymmetric galaxy

are like Lissajous ﬁgures. In his book (Contopoulos, 2002), Contopoulos explains

that in 1958 he expected the orbits to be ergodic and ﬁll all the space inside the

energy surface. Instead, he found that the orbits did not ﬁll all the available space,

but instead ﬁlled curvilinear parallelograms, as in the case of deformed Lissajous

ﬁgures. Later he could prove (Contopoulos, 1960) that such orbits can be explained

both qualitatively and quantitatively by a formal third integral of motion.

The work of H

enon and Heiles (1964) on galactic motion is the ﬁrst complete work

that establishes the existence of chaos in galaxy formations. It provided considerable

evidence of the existence of chaotic motions in galaxies. Their work started as a

computer experiment in order to explore the existence and applicability of the third

integral of galactic motion. Chapter 12 is devoted to this work of H

enon-Heiles and

Contopoulos.

The works of Contopoulos and H

enon-Heiles are examples of what we will call

computer experiments. This new type of experiments gave new directions in various

scientiﬁc ﬁelds, and especially in astronomy. Sometimes the computer results were

surprising and contradicted the existing theories of that time.