0001
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`Luminara 2039
`Liown v. Disney
`IPR2015-01656
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`

`
`o
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`WIT
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`SD
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`P
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`co
`4-40
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`4.5441
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`I \I
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`tAn
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`0002
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`

`
`Contents
`
`List of colour plates
`Preface
`Acknowledgements
`How to read the book
`
`page ix
`xiii
`xv
`xvi
`
`Part I The phenomenon: complex motion,
`unusual geometry
`
`1 Chaotic motion
`1.1 What is chaos?
`1.2 Examples of chaotic motion
`1.3 Phase space
`1.4 Definition of chaos; a summary
`1.5 How should chaotic motion be examined?
`Box 1.1 Brief history of chaos
`
`2 Fractal objects
`2.1 What is a fractal?
`2.2 Types of fractals
`2.3 Fractal distributions
`2.4 Fractals and chaos
`Box 2.1 Brief history of fractals
`
`Part II
`
`Introductory concepts
`
`3 Regular motion
`3.1
`Instability and stability
`Box 3.1
`Instability, randomness and chaos
`3.2 Stability analysis
`3.3 Emergence of instability
`Box 3.2 How to determine manifolds numerically
`3.4 Stationary periodic motion: the limit cycle (skiing
`on a slope)
`3.5 General phase space
`
`3
`
`3
`4
`19
`21
`22
`23
`
`24
`24
`32
`40
`45
`47
`
`49
`
`51
`51
`59
`65
`67
`73
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`76
`79
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`90
`4 Driven motion
`90
`4.1 General properties
`95
`4.2 Harmonically driven motion around a stable state
`98
`4.3 Harmonically driven motion around an unstable state
`100
`4.4 Kicked harmonic oscillator
`4.5 Fixed points and their stability in two -dimensional maps 103
`105
`4.6 The area contraction rate
`4.7 General properties of maps related to differential
`equations
`Box 4.1 The world of non - invertible maps
`In what systems can we expect chaotic behaviour?
`Investigation of chaotic motion
`
`106
`108
`109
`
`111
`
`4.8
`
`Part III
`
`5 Chaos in dissipative systems
`5.1 Baker map
`5.2 Kicked oscillators
`Box 5.1 Hénon -type maps
`5.3 Parameter dependence: the period- doubling cascade
`5.4 General properties of chaotic motion
`Box 5.2 The trap of the `butterfly effect'
`Box 5.3 Determinism and chaos
`5.5 Summary of the properties of dissipative chaos
`Box 5.4 What use is numerical simulation?
`Box 5.5 Ball bouncing on a vibrating plate
`5.6 Continuous -time systems
`5.7 The water -wheel
`Box 5.6 The Lorenz model
`
`113
`114
`131
`147
`149
`154
`159
`168
`171
`172
`174
`175
`181
`187
`
`191
`193
`199
`
`6 Transient chaos in dissipative systems
`6.1 The open baker map
`6.2 Kicked oscillators
`Box 6.1 How do we determine the saddle
`201
`and its manifolds?
`202
`6.3 General properties of chaotic transients
`210
`6.4 Summary of the properties of transient chaos
`211
`Box 6.2 Significance of the unstable manifold
`213
`Box 6.3 The horseshoe map
`214
`6.5 Parameter dependence: crisis
`217
`6.6 Transient chaos in water -wheel dynamics
`219
`6.7 Other types of crises, periodic windows
`221
`6.8 Fractal basin boundaries
`225
`Box 6.4 Other aspects of chaotic transients
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`Contents
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`vii
`
`7 Chaos in conservative systems
`7.1 Phase space of conservative systems
`7.2 The area preserving baker map
`Box 7.1 The origin of the baker map
`7.3 Kicked rotator - the standard map
`Box 7.2 Connection between maps and
`differential equations
`Box 7.3 Chaotic diffusion
`7.4 Autonomous conservative systems
`7.5 General properties of conservative chaos
`7.6 Summary of the properties of conservative chaos
`7.7 Homogeneously chaotic systems
`Box 7.4 Ergodicity and mixing
`Box 7.5 Conservative chaos and irreversibility
`
`8 Chaotic scattering
`8.1 The scattering function
`8.2 Scattering on discs
`8.3 Scattering in other systems
`Box 8.1 Chemical reactions as chaotic scattering
`8.4 Summary of the properties of
`chaotic scattering
`
`9 Applications of chaos
`9.1 Spacecraft and planets: the three -body problem
`Box 9.1 Chaos in the Solar System
`9.2 Rotating rigid bodies: the spinning top
`Box 9.2 Chaos in engineering practice
`9.3 Climate variability and climatic change: Lorenz's
`model of global atmospheric circulation
`Box 9.3 Chaos in different sciences
`Box 9.4 Controlling chaos
`9.4 Vortices, advection and pollution: chaos in fluid flows
`Box 9.5 Environmental significance of
`chaotic advection
`
`10 Epilogue: outlook
`Box 10.1 Turbulence and spatio- temporal chaos
`
`Appendix
`A.1 Deriving stroboscopic maps
`A.2 Writing equations in dimensionless forms
`
`227
`227
`230
`233
`234
`
`236
`239
`242
`250
`259
`260
`261
`262
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`264
`265
`266
`274
`276
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`277
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`279
`279
`284
`285
`292
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`293
`300
`303
`304
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`315
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`3.18
`320
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`322
`322
`325
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`A.3 Numerical solution of ordinary differential equations
`AA Sample programs
`A.5 Numerical determination of chaos parameters
`
`Solutions to the problems
`Bibliography
`Index
`
`329
`332
`337
`
`342
`370
`387
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`3
`
`and specific tools. Traditional methods are unsuitable for the description
`Understanding chaotic motion requires a non -traditional approach
`
`mine whether or not the motion will he regular.
`accepted view, the simplicity of the equations of motion does not deter-
`have very complicated solutions. Contrary to the previously generally
`tence of such motion is due to the fact that even simple equations can
`a few components, is called chaotic. As will be seen later, the exis-
`The irregular motion of simple systems, i.e. systems containing only
`
`sort can be recognised in the dynamics.
`simple superposition of swinging and oscillation. No regularity of any
`good example: for large amplitudes it is much more complex than the
`itself. The motion of a body fastened to the end of a rubber thread is a
`motion, even of simple systems, is often irregular and does not repeat
`sustained motion. It has become widely recognised that long -lasting
`Regular motion, however, forms only a small part of all possible
`
`dynamics when the return velocity is plotted against the position.
`tion with exactly the same velocity, i.e. a single point characterises the
`is suitable for measuring time); (3) it always returns to a specific posi-
`state is accurately predictable (this is precisely why a pendulum clock
`characteristics of a periodic motion arc: (I) it repeats itself; (2) its later
`most superposition of periodic motion with different periods). Important
`tional education, sustained motion is always regular, i.e. periodic (or at
`the Earth orbiting the Sun. According to the view suggested by conven-
`Examples from everyday life are the swinging of a pendulum clock or
`Certain long- lasting, sustained motion repeats itself exactly, periodically.
`1.1 What is chaos?
`
`Chaotic motion
`Chapter 1
`
`0007
`
`

`
`Semions 5.6.2 and 5.6.3.
`I The precise equations of motion of the examples in this section can be found in
`
`statistical sense.
`of a similar nature in both cases: the two motions are equivalent in a
`unpredictable. This figure also shows that the long -term behaviour is
`in the displacement after only a short time (Fig. 1.3): the dynamics is
`Slightly different initial conditions result in significant differences
`repetition in the displacement vs. time curve; i.e., the motion is irregular.
`spring and driven sinusoidally) It can clearly be seen that there is no
`Figure 1.2 shows the motion of a body fixed to the end of a stiffening
`
`not repeat itself regularly; it is chaotic.
`the amplitude nor the frequency is uniform: the sustained motion does
`spin -dryer, an irregular sound can be heard in such situations). Neither
`differs from it in detail (instead of the uniform hum of the car or the
`dynamics follows the driving force in an averaged sense only, but always
`its own periodic behaviour is no longer harmonic. Thus, the sustained
`exactly the sinusoidal, harmonic motion of the forcing apparatus, since
`be chaotic. A qualitative explanation is that the spring is not able to adopt
`non -linearity is involved, the sustained state of the driven oscillation may
`speak of stiffening or softening springs, respectively. Whichever type of
`ormore slowly than it would in linear proportion to the elongation: we can
`For non -linear force laws, the restoring force increases more rapidly
`
`non -linearity.
`Resonance is therefore a characteristic example for the appearance of
`longer proportional to the elongation; i.e., the force law is non -linear.
`ops. For large amplitudes, however, the force of the spring is usually no
`become very large and the well known phenomenon of resonance devel-
`the spring is close to that of the driving force, then the amplitude may
`ular: it adopts the period of the driving force. If the natural period of
`proportional to the elongation. In this case the sustained motion is reg-
`
`presence of friction.
`stiffening spring (a driven non- Ilnear oscillator), driven sinusoidally in the
`Flg. 1.2. Irregular sustained oscillations of a point mass fixed to the end of a
`
`law to a good approximation: the magnitude of the restoring force is
`As long as the displacement is small, the spring obeys a linear force
`
`Fig. 1.1.
`distribution of clothes in the spin- dryer), as indicated schematically in
`interactions with pot -holes in the case of the car wheel and by the uneven
`less periodic shaking, i.e, the application of a driving force (caused by
`supplied from an external source. The supplied energy can be a more or
`and ultimately vanish. Sustained motion can only develop if energy is
`to friction or air drag, these oscillations, when left alone, are damped
`spin- dryers) oscillate. Because of the losses that are always present due
`Objects mounted on spring suspensions (for example car wheels and
`
`Irregular oscillations, driven pendulum - the
`
`chaotic attractor
`1.2.1
`1.2 Examples of chaotic motion
`
`detailed understanding of chaotic dynamics.
`types of chaos and help in developing the new concepts necessary for a
`tially different behaviour. These examples also serve to classify different
`that slightly different choices of the parameters could result in substan-
`that all of our examples are discussed for a unique set of parameters, and
`which are unavoidable when studying chaos. It should be emphasised
`motion of very simple systems on the basis of numerical simulations,
`considering particular cases. In the following, we present the chaotic
`vidually or together; the most efficient way to understand them is by
`The properties of chaotic systems am unusual, either taken indi-
`
`dynamics are summarised in Table 1.1.
`vs. velocity representation. The differences between the two types of
`complicated: a complex but regular structure appears in the position
`initial conditions that arc never exactly known, (3) the return rule is
`not repeat itself, (2) it is unpredictable because of its sensitivity to the
`by the opposite of the three properties mentioned above: (I) it does
`observations have led to the result that chaotic motion is characterised
`has become possible through computer -based experimentation. Detailed
`of such motion, and the discovery of the ubiquity of chaotic dynamics
`
`of complicated geometry
`unpredictable
`irregular
`
`Chaotic motion
`
`of simple geometry
`predictable
`self -repeating
`
`Regular motion
`
`Table 1.1. Comparison of regular and chaotic motion.
`
`moved sinusoidally with time.
`other end of the spring is
`weightless spring and the
`mass is fixed ta one end of a
`oscillations: a body of finite
`Fig. 1.1. Model of driven
`
`5
`
`1 Chaotic motion
`
`4 The phenomenon: complex motion, unusual geometry
`
`0008
`
`

`
`horizontal plane.
`point of suspension in the
`periodic movement of its
`the pendulum is driven by the
`Fig. 1.5. Driven pendulum:
`
`that will therefore be called an attracting object, or an attractor (for the
`the initial state, the dynamics converges to some sustained behaviour
`velop if some external energy supply (driving) is present. Regardless of
`In a frictional (dissipative) system, sustained motion can only de-
`
`ing to the period of the driving force (Fig. 1.8 and Plate II).
`velocity (angular velocity) co- ordinates (x,,, v ) at intervals correspond-
`taking samples from it by plotting the position (angular deflection) and
`strated by following the motion initiated in Fig. 1.6 for a long time and
`The structure underlying the irregular motion can again be demon-
`
`series of unstable states.
`The reason for the unpredictability is that the motion passes through a
`over, while the other one falls back to the side it came from (Fig. 1.7).
`state, an 'upside down' state, separates them. Then one of them turns
`nearby initial positions remain close to each other only until an unstable
`a pencil standing on its point. Two paths of the pendulum starting from
`motion. The `upside down' state is especially unstable, just like that of
`Note that the pendulum turns over several times in the course of its
`
`of the pendulum in the vertical plane.
`driving force, the motion may become chaotic. Figure 1.6 shows the path
`considered to be fixed to a very light, thin rod. With a sufficiently
`order to avoid the problem of the folding of the thread, the point mass is
`the point of suspension is moved horizontally, sinusoidally in time. In
`The pendulumcan be driven in different ways. We examine thecase when
`ceases because of friction or air drag: sustained motion is impossible.
`but to the sine of this angle. Without any driving force, the swinging
`linear, since the restoring force is not proportional to the deflection angle
`The large -amplitude swinging of a traditional simple pendulum is non-
`Another example is the behaviour of a driven pendulum (Fig. 1.5).
`
`suspension point moves.
`and often turns over. The horizontal bar indicates the interval over which the
`end -point of the pendulum for a longer time: the pendulum swings irregularly
`starting from a hanging stale (over the first half period). (b) The path of the
`Fig. 1.6. Motion of a driven pendulum. (a) The pendulum a few moments after
`
`(b)
`
`(a)
`
`therefore infinitely more complicated than periodic motion.
`respond to a periodic motion in this representation. Chaotic motion is
`will be given in Chapter 2). Remember that a single point would cor-
`objects: it is a structure called a fraclal (a detailed definition of fractals
`is much more complicated than those of traditional plane -geometrical
`indicating that chaos is associated with a definite structure. This pattern
`interval anywhere. The whole picture has a thready, filamentary pattern,
`sponding to a single position co- ordinate x do not form a continuous
`velocity values belong. Furthermore, the possible velocity values corre-
`(according to detailed examinations, an infinite number of) different
`It is surprising that there are numerous values of x, to which many
`
`periods.
`ples, n, of the period of the driving force, through several thousands of'
`velocity co- ordinates (x,,, u) of the sustained motion at integer multi-
`Figure 1.4 and Plate I have been generated by plotting the position and
`tion continuously, but only 'take samples' of it at equal time intervals.
`An interesting structure reveals itself when we do not follow the mo-
`
`0
`-50
`
`o
`
`lln
`
`50
`
`s
`
`are x,, and v,,, respectively.
`co-ordinates of the nth sample
`position and velocity
`of the driving force. The
`corresponding to the period
`samples taken at time intervals
`position representation, using
`oscillation in the velocity vs.
`from a sustained non -linear
`Fig. 1.4. Pattern resulting
`
`therefore it is unpredictable.
`to the initial conditions and
`rapidly: the motion is sensitive
`initial difference increases
`identical positions. The small
`which started from nearly
`Fig. 1.3. Two sets of motion
`
`7
`
`1 Chaotic motion
`
`6 The phenomenon: complex motion, unusual geometry
`
`r
`
`0009
`
`

`
`2 The equations of morion of the magnetic pendulum can be found in Section 6.8.3.
`
`i
`
`can again be obtained by representing the starting point in the position
`tractors in the vertical plane). An overall view of the basins of attraction
`Fig. 1.12, which depicts the paths corresponding to these two simple at-
`tained motion only. There are two options for the given parameters (see
`the friction is sufficiently large, the pendulum can exhibit regular sus-
`A drivenpendulum (Fig. 1.5) may also exhibit transient chaos. When
`
`time (Fig. 1.11), but ultimately it ends up on one of the at-tractors.
`while, exhibiting nansien( chaos, i.e. chaos lasting for a finite period of
`Motion starting near the fractal boundary remains irregular for a
`
`attractors.)
`appears in one colour only: the boundaries do not come close to the
`fractal basin boundaries. (Naturally, the close vicinity of each attractor
`manner (see Fig. 1.10 and Plates III -VI); these simple attractors have
`the basin boundaries are interwoven and entangled in a complicated
`Each identically coloured area is a basin of attraction. Surprisingly,
`tions that converge towards them, the whole plane can be coloured.
`colours to the three attractors, and to the corresponding initial posi-
`will be closest to after coming to rest'- By assigning three different
`plane, we can use a computer to calculate which magnet the pendulum
`three simple attractors in the system. Starting above any point of the
`come to a halt, pointing towards any of the magnets. Thus there are
`end of the pendulum and the magnets is attracting. the pendulum can
`
`horizontal equilateral triangle (Fig. 1.9). When the force between the
`body, moving above three identical magnets placed at the vertices of a
`Consider a pendulum, the end -point of which is a small magnetic
`boundary - transient chaos
`1.2.2 Magnetic and driven pendulums, fractal basin
`
`attractors.
`of its peculiar structure. Figures 1.4 and 1.8 display examples of chaotic
`presence of a chaotic attractor, also called a strange attractor because
`namics is then usually irregular, i.e. chaotic. This is accompanied by the
`inevitably brings about the non- linearity of the system; the sustained dy-
`to regular or to ceasing motion. A sufficiently large supply of energy
`exact definition, see Section 3.1.2). Simple attractors correspond either
`
`irregular, chaotic motion.
`magnets, but only after some
`pointing towards one of the
`settles in an equilibrium state
`The pendulum ultimately
`fixed to the end of the thread.
`attracted by the magnets is
`to the table and a point mass
`pendulum: magnets are fixed
`FIg. 1.9. The magnetic
`
`velocity.
`point with zero initial
`when starting above that
`pendulum comes to a rest
`whose neighbourhood the
`according to the magnet in
`horizontal plane is shaded
`Each point on the
`white and two black dots).
`magnetic pendulum (one
`equilibrium states of the
`attraction of the three
`Fig. 1.10. Basin of
`
`driving period:
`at integer multiples of the
`position -velocity co- ordinates
`of the pendulum in the
`obtained by plotting the state
`pendulum (chaotic attractor)
`from a chaotic driven
`Fig. 1.8. Pattern resulting
`
`pendulums move.
`which the end -points of the
`arrows show the direction in
`the same as in Fig. 1.6. The
`unstable state. The notation is
`nearby points while passing an
`pendulums starting from
`paths of two identical driven
`Fig. 1.7. Separation of the
`
`9
`
`1 Chaotic motion
`
`B The phenomenon: complex motion, unusual geometry
`
`r
`
`0010
`
`

`
`dynamics of transient type.
`responsible for chaotic
`This chaotic saddle is
`period of the driving force.
`themselves after every
`moving between
`followed in time, they keep
`the basin boundary and, if
`points shown here are on
`either simple attracton all
`Fig. 1.13 that never reach
`the driven pendulum of
`Fig. 1.14. Initial states of
`
`white, respectively.
`are marked in black and
`converging towards them
`and the initial states
`(white and black dots),
`1.12 appear here as points
`simple attractors in Fig.
`initial conditions. The two
`pendulum on the plane of
`attraction in the driven
`Fig. 1.13. Basins of
`
`fractal curves. The motion starting from the vicinity of these fractal basin
`often penetrate each other, and their boundaries can also be filamentary
`conditions whichconverges to the given attractor. Thebasins of attraction
`exist, each with its own basin of attraction defined by the set of initial
`transients) leading to them. In such cases several simple attractors co-
`motion are regular, but there are many possible transient routes (chaotic
`Thus, chaotic dynamics can also occur if the sustained forms of
`
`chaotic saddle.
`domain in the plane, but rather a fractal cloud of isolated points called a
`but the initial conditions that describe these state do not form a compact
`any length of time. There exists an infinity of such motion (Fig. 1.14),
`which the dynamics never reaches any of the attractors, and is chaotic for
`transient. There exist, however, very exceptional initial conditions from
`of the simple attractors. Irregular dynamics has a finite duration; it is
`in the case of the chaotic attractor, but it ultimately converges to one
`Motion starting close to the boundary is similar initially to that seen
`Plate VII).
`the attractor which the motion ultimately converges to (Fig. 1.13 and
`(angular deflection) - velocity (angular velocity) plane in the colour of
`
`attractor each.
`initial conditions lead to one of these motions, corresponding to a simple
`strong friction only these two types of sustained motion exist. All the different
`Fig. 1.12. Simple periodic attractors of the driven pendulum: for sufficiently
`
`o
`
`-1
`
`o
`
`y
`
`dots.)
`are represented by solid black
`chaotic. (The fixed magnets
`rest positions: it is transiently
`before reaching one of the
`above. The motion is irregular
`pendulum viewed from
`end -point of the magnetic
`Fig. 1.11. Path el the
`
`11
`
`1 Chaotic motion
`
`10 The phenomenon: complex motion, unusual geometry
`
`0011
`
`

`
`swinging thread on time within the same time interval.
`pulley by the centre of an open circle); (b) the dependence of the length of the
`path of the swinging body (the initial position is marked by a black dot, the
`Fig. 1.16. Frictionless motion of a body swinging on a pulley. (a) The spa ai
`
`t
`11-
`
`50
`
`40
`
`0
`
`20
`
`10
`
`0
`
`1
`
`(b)
`
`0.2
`
`0.4
`
`0.6
`
`0.8 -
`Il
`
`13
`
`1 Chaotic motion
`
`12 The phenomenon: complex motion, unusual geometry
`
`and 7.4.3.
`3 The equations of motion of the examples in this section can be found in Sections 7.4.1
`
`conditions soon branch off; the motion is unpredictable.
`spectively. Again, the paths of the motion starting from nearby initial
`length of the thread vs. time are shown in Figs. 1.16(a) and (b), re-
`chaotic motion may develop. The path of the swinging body and the
`from the pulley when turning over.) Thus, a long- lasting, complicated,
`collide with anything and that the thread does not become unattached
`if the latter is the heavier. (It is assumed that the swinging body does not
`spins faster, and thus becomes able to pull the other body upward, even
`the swinging body turns over several limes, the thread shortens, the body
`with sufficient momentum while the other body moves downwards, then
`is much more interesting now. If the swinging body is thrust horizontally
`allowed, the heavier mass always pulls the other one up, but the situation
`tion. In the traditional arrangement, where only vertical displacement is
`swing is one of the position co- ordinates; the other is the angle of deflec-
`The instantaneous length, I, of the thread of the point mass that can
`
`called the chaotic band (Fig. 1.17). Other initial conditions outside of the
`in a disordered manner and dotting a finite region of the plane; This is
`chaotic motion is represented by a sequence of points jumping around
`change of this length, u , will be plotted as one point in the plane. Thus,
`the instantaneous length, l e x , of the swinging thread and the rate of
`the swinging body passes through the vertically hanging configuration,
`at identical time intervals, rather at identical configurations: whenever
`is not driven in this case, and therefore sampling will not take place
`can be presented with the help of some sampling technique. The system
`An overview of the motion corresponding to a given total energy
`
`motion develop under such conditions.3
`for the sake of simplicity). It will be shown that new types of chaotic
`point masses swing in a vertical plane (with the thread always stretched
`well known secondary school problem. Here, however, we let one of the
`(see Fig. 1.I5). The case when both points can only move vertically is a
`sider two point masses joined by a thread wound about a small pulley
`Let us examine what happens in frictionless (conservative) systems. Con-
`on slopes - chaotic bands
`1.2.3 Body swinging on a pulley, ball bouncing
`
`uncertainty the motion is irregular and is bound to fractal structures.
`it is difficult to decide which attractor to choose. During this period of
`boundaries behaves randomly along the boundary for a long time, as if
`
`other moving vertically only.
`freely in a vertical plane, the
`radius, one of them swinging
`around a pulley of negligible
`joined by a thread wound
`a pulley: two point masses are
`Fig. 1.15. Body swinging on
`
`Plate VIII).
`texture, different from the fractals presented so far (sec Fig. 1.17 and
`and islands. Together they form a complicated structure of interesting
`systernis characterisedby a hierarchically nestedpattern of chaotic bands
`closed domains that can be called regular islands. A frictionless chaotic
`which correspond to regular motion. These objects together usually form
`band may result in a single point, a few points or a continuous line, all of
`
`surfaces can also lead to chaotic motion. Maybe the simplest situation
`Our second example illustrates the fact that elastic collisions with flat
`
`islands.
`closed curves form regular
`condition. The sets of
`starting from a single initial
`be traced out by motion
`chaotic band, which can
`left. The dotted region is a
`vertical position, from the
`when passing through the
`and velocity (x, tut taken
`basis of samples of length
`given total energy, on the
`without air drag and at e
`swinging on a pulley
`the motion of a body
`Fig. 1.17. Overview of
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`4 A detailed investigation of this problem can be found in Section 8.2.3.
`
`many collisions (Fig. 1.22). Moreover, there is an infinity of initial points
`strongly depends on the initial position. Some initial conditions lead to
`particles that start along a segment in a given direction towards the discs
`fests itself in several ways. The number of bounces experienced by the
`each collision. The complicated structure related to the motion mani-
`and the angle between the straight sections of the paths increases with
`conditions is easy to explain, since the discs act as dispersing mirrors
`this motion is also chaotic. The deviation of paths with nearby initial
`tial conditions cause the paths to diverge rapidly (Fig. 1.21); therefore,
`of the particle is complicated and aperiodic.4 'Wm slightly different ini-
`for a very long time between the discs; during this time the dynamics
`the velocity vector. Some initial conditions cause the particle to bounce
`Starting from a givenpoint, the motion depends on the initial direction of
`therefore the velocity of the particle is constant during the entire process.
`ball machine (sec Fig. 1.21). The motion is considered to be frictionless;
`the horizontal plane and a ball is bouncing among them - like in a pin-
`Three identical discs are placed at the vertices of a regular triangle in
`Christmas -tree ornaments - chaotic scattering
`1.2.4 Ball bouncing between discs, mirroring
`
`bands that, contrary to chaotic attractors, are plane- filling objects.
`sets. The initial conditions that lead to chaotic motion forrn chaotic
`tain sets of finit al conditions, while chaotic motion corresponds to other
`initial condition s and the total energy. Regular motion corresponds to cer-
`systems. As a esult, the nature of all motion strongly depends on the
`
`soon diverge.
`from nearby initial points
`triangle. Paths starting
`vertices of a regular
`identical discs fixed at the
`elastically between
`bouncing perfectly
`problem: particles
`Fig. 1.21. The three -disc
`
`tained motion because there are no attractors in frictionless, conservative
`On the other hand, this motion cannot converge to a well defined sus-
`frictionless systems, since there is no dissipation and energy is conserved.
`There is no need to apply driving forces in order to sustainamotion in
`
`at the instant of each bounce (Fig. 1.20).
`is in this case to plot the two velocity components as points of a plane,
`A sampling technique providing a good overview of the dynamics
`
`as in the previous examples.
`heights but slightly different positions soon branches off (Fig. 1.19), just
`tween the slopes. The chaotic motion of two balls dropped from identical
`Non -linearity and inherent instability are caused by the break -point be-
`the opposite slope the ball does not necessarily hit its original position.
`with atoms.) Chaotic behaviour arises because after bouncing back from
`(Fig. 1.18). (A motion very similar to this can be realised in experiments
`is the case of an elastic ball bouncing on two slopes that face each other
`
`slope is 50 °.
`angle of inclination of the
`region Is a chaotic band. The
`nth bounce. The dotted
`taken at the instance of the
`perpendicular to the slope (z)
`the square of the component
`slope (u) and the ordinate is
`component parallel to the
`abscissa is the velocity
`in a representation where the
`slope with given total energy
`ball bouncing on a double
`by the possible mations of a
`Fig. 1.20. Pattern generated
`
`conditions.
`sensitive to the Initial
`In Fig. 1.18). The motion is
`line is identical to that drawn
`double slope (the continuous
`initial positions above the
`starting from nearly Identical
`Fig. 1.19. Paths of two bails
`
`other in a gravitational field.
`inclination that face each
`two slopes of identical
`Fig. 1.18. Ball bouncing on
`
`15
`
`1 Chaotic motion
`
`14 The phenomenon: complex motion, unusual geometry
`
`r
`
`0013
`
`

`
`5 The equation of motion for this example can he found in Section 9.4.1.
`
`surfaces through the mixing of cream in coffee to the propagation of
`be observed in numerous phenomena, ranging from oil stains on road
`The spreading of impurities in the form of filamentary patterns can
`
`Plate Xl).
`losing its original compact shape) within a short time (Fig. 1.25 and
`cle, the drop traces out a well defined thready fractal structure (after
`discovery is that, despite the chaotic motion of each individual parti-
`a certain initial region, the initial shape of the droplet. A surprising
`nation of the dynamics of an ensemble of particles, each starting from
`to follow the motion of a dye droplet. This corresponds to the exami-
`In the context of the spreading of pollution, it is especially important
`
`close to each other, but leaving the tank via different outlets.
`the container. Figure 1.24 illustrates two complicated paths starting very
`so on. It may thus take a very long time before the particle flows out of
`towards the other outlet, but again it may be too late to be drained, and
`outlet may not reach it within half a period; therefore it starts moving
`easy to follow.5 Chaos arises because a particle moving towards the open
`particle is identical to that of the liquid. The path of the particle is then
`is determined by the condition that the instantaneous velocity of the
`of the liquid; the only difference is the colour. In this case, the motion
`it is assumed that the material properties of the dye are identical to that
`know how a dye particle moves in this flow. For the sake of simplicity,
`half a period (Fig. 1.23), yielding a flow periodic in time. We want to
`whirls while flowing out. The two outlets are alternately open, each for
`Consider a large and flat container with two point -like outlets. Water
`
`this matter is obvious.
`in a flowing medium (air or water). The environmental significance of
`
`advection of the particles is nevertheless chaotic.
`the first and second half period, respectively. The flow itself is very simple; the
`generate chaotic advection in a flat container. (a) and (b) illustrate the flow in
`Fig. 1.23. Tank with two outlets. The outlets, when opened alternately,
`
`1;3
`
`applications. One of these is discussed here: the spreading of pollutants
`Chaotic motion occurs in numerous phenomena related to practical
`1.2.5 Spreading of pollutants - an application of chaos
`
`transient chaos.
`similar to that of simple attractors.