oscillations - uni-potsdam.de · 2.3 simple harmonic oscillator ... had the idea that any...

Oscillations

Kai Hoffmann, Tobias Jurk, Albrecht Kroner,Hannes Matuschek, Gregor Monke, Denis Ratzel,

Philipp Thomas

2006-04-28

1

CONTENTS

Contents

1 Historical introduction 3

1.1 Mathematical basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Physical basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Introduction to oscillations 5

2.1 Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Period and frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Simple harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Conservative systems 7

3.1 Definition and discription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 The harmonic oszillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 The double pedulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3.1 Numerical solution for small elongations of the double pendulum . . . . . . . . . . 8

3.3.2 Analytic solution for small elongations of the double pendulum . . . . . . . . . . . 9

3.3.3 Numerical solutions for larger elongations of the double pendulum . . . . . . . . . 9

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Forced oscillations 11

4.1 Unforced damped oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Experiment 1: Angular frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3 Experiment 2: Damping factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.4 Forced damped oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.5 Experiment 3: Amplification factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.6 Experiment 4: Phase shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.7 Experiment 5: Unforced damped harmonic oscillations . . . . . . . . . . . . . . . . . . . . 14

4.8 Experiment 6: Chaotic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Numerical analysis 15

1

CONTENTS

6 The Lorenz system 19

6.1 Intro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2 Physical meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.3 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.3.1 State of no convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.3.2 Stationary convection rolls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.3.3 Chaotic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.3.4 Coexistence of chaotic and nonchaotic behaviour . . . . . . . . . . . . . . . . . . . 22

6.3.5 Period doubling route to chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7 Self-sustained systems in electrical engineering 24

7.1 Charging and discharging a capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7.1.1 Technical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.2 Relaxation oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.3 Properties of alternating currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7.3.1 High- and low-pass filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Van der Pol Oscillator 33

8.1 Definition and discription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8.2 Van der Pol oscillator as a harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 34

8.3 Van der Pol oscillator as a self-sustained system . . . . . . . . . . . . . . . . . . . . . . . . 36

8.4 Relaxation oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.5 Van der Pol oscillator with hard excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.6 Synchronisation of the Van der Pol oscillator by an external force . . . . . . . . . . . . . . 39

9 Outlook: Synchronizing chaos 43

2

1 HISTORICAL INTRODUCTION

1 Historical introduction

1.1 Mathematical basics

The first known accurate definition of sine was given by Mohammed ibn Dschabir al-Battani (858-929),who based his ideas on greek and indian knowledge. Though complex numbers only became importantin the 18th century, the earliest fleeting reference to square roots of negative numbers probably occurredin the work of Heron of Alexandria (c. 10-70), a respected constructer and mathematician in Egypt.

Muhammed ibn Musa Alcharizmi (c. 780-850), who worked in Baghdad almost at the same time asDschabir al-Battani, pointed out the problem of taking square roots of negative numbers.

Independent from Musa Alcharizmi, the indian mathematician Madhava of Sangamagrama (1350-1425)gave an approximation of transcendental numbers (non-algebraic complex numbers) by continued frac-tions. Obviously this knowledge was not distributed because 100 years later Gerolamo Cardano (1501-1576) formulated the problem of imaginary numbers again and probably did the first algebraic calculationsusing complex numbers.

Raffaele Bombelli (1526-1573) gave essential contributions to the theory of complex numbers. The methodof doing calculations using the complex plane was established by Pierre-Simon Laplace (1749-1827) andLeonard Euler (1707-1783), but in an intuitive way without a clear mathematical prove.

Louis Cauchy (1789-1857) defined accurately complex functions using the idea of the complex plane andfound various useful applications, for example the residuum theory. Probably the most important theoryfor describing oscillations is the infinitesimal calculus, which was developed by Isaac Newton (1643-1724)and Gottfried Wilhelm Leibnitz (1646-1716) independent from each other.

Jean Baptiste Fourier (1768-1830) had the idea that any oscillation is just a superposition of manyharmonic oscillations known as the Fourier theorem necessary for every analysis of any oscillation.

1.2 Physical basics

Mechanics Robert Hooke (1638-1703) was the first one, who found the complicate relation betweenamplitude and force of a spring.

Light waves Describing waves became interesting for physicists since Christiaan Huygens (1629- 1695)investigated the libration and wavelike character of light. Interference effects of light waves where laterproved by Thomas Young (1773-1829).

Acoustics The first intensive analysis of propagation and speed of acoustic waves was done by ErnstChladni (1756-1827). Georg Simon Ohm (1789-1854) identified tone as a superposition of basic tints.Supersonic effects where investigated by Ernst Mach (1838-1916) observing very fast bullets.

Classical electro dynamics The theory of classical electrodynamics was completed by James ClarkMaxwell (1831-1879). His four equations can be used to describe electro-magnetic waves. The firstartificial generation of electro-magnetic waves was done by Frank Hertz (1857-1894).

3

1 HISTORICAL INTRODUCTION

quantum mechanics Albert Einstein explained the photo-electric effect with a wave-particle dual-ism of light. Louis-Victor de Broglie postulated the wave-particle-dualism (1892-1987) of every matter.Erwin Schrodinger (1887-1961) describes quantum particle in a potential field as a time-dependent wave-equation.

Chaotic research In 1960 Edward N. Lorenz discovered the deterministic Chaos. 1970/1980 MitchellFeigenbaum developed logical equations, the Feigenbaum constant and the Feigenbaum diagramm.

4

2 INTRODUCTION TO OSCILLATIONS

2 Introduction to oscillations

How we can determine an oscillating system in an easy and short way? In such systems a particle or aset of particles moves back and forth. Whether it be a ball bouncing on a floor, a pendulum swingingback and forth, or a spring compressing and stretching, the basic principle of oscillation maintains thatan oscillating particle returns to its initial state after a certain period of time. So we are dealing withperiodic motions, and those can be found in all areas of physics. To define it a little more precisely, wecan look at the forces which are acting on an oscillating particle.

At first we have to keep in mind that in every oscillating system there is an equilibrium point at whichno resultant force acts on the particle. For instance, in the case of a spring with a mass the state ofequilibrium is the point where the spring is relaxed. If we define the state of equilibrium as x = 0, we cangenerally say, that any deviation from that point causes a force, which always acts in a direction oppositeto the displacement of the particle. This force is called restoring force. After an initial pertubation, andas long as the force obeys the above principle,the resulting motion is oscillatory. Oscillations can be verycomplex, but for this small introduction, we focus on free harmonic oscillations. To do so, we shouldintroduce some variables.

Of course, as in classical study of motion, we need x and x. But dealing with periodic motion threatsalso amplitude and phase.

2.1 Amplitude

A simple oscillator generally goes back and forth between two extreme points; the points of maximumdisplacement from the equilibrium point. We shall denote this point by xm and define it as the amplitudeof the oscillation. If a pendulum is displaced 1cm from equilibrium and then allowed to oscillate we cansay that the amplitude of oscillation is 1cm.

2.2 Period and frequency

In simple oscillations, a particle takes a specific time to return to its initial position, this time is calledthe period T . Related to the period, the frequency f yields the number of cycles per unit time, and isso for given by f = 1

T . Period is measured in seconds, while the unit of frequency is Hertz (Hz), where1 cycle

second = 1Hz. Not only for circular motions we can introduce the angular frequency ω, which definesthe number of radians per second, and is given by ω = 2 π

T . Oscillating systems do complete cycles andif we think of each cycle as containing 2π radians, then we can define angular frequency. Now we areprepared for the basic model of oscillations.

2.3 Simple harmonic oscillator

We start with a common example, a mass on a spring. The force acting on the mass depends on theproperty of the spring k, and the deviation of the mass x. On every moment the restoring force, inthis example given by the elastic deformation of the spring, tries to push the mass back to the point ofequilibrium. If we apply Hook’s law we get:

F = −k(x− x0) (1)

5

2 INTRODUCTION TO OSCILLATIONS

Of course here we leave out any friction or distension, because we want a simple linear model. Now weuse Newton’s second law and maintain (x0 = 0) for equation (1):

ma = −kx (2)

This is differential equation for x(t), rewritten in terms of derivatives:

0 = x + ω2x (3)

ω =

√k

m(4)

Equation (3) has a partial solution:x(t) = xm · cos (ωt) (5)

This yields a harmonic oscillation with constant angular frequency and amplitude. For the period we get:

T = 2π

√m

k

We obtain, that the angular frequency depends only on the spring constant k and the mass m. Theinitial displacement, the amplitude, is completely independent of the characteristics of the motion. Thisconcept is very important and is valid for every problem that can be modeled as a harmonic oscillator.

6

3 CONSERVATIVE SYSTEMS

3 Conservative systems

3.1 Definition and discription

Systems that preserve their energy are called conservative systems. In such a system the phase volumeis preserved, too. Most cases a system can only be discribed approximately as a conservative system.The differetial equation that discribe such a system can be found with Hamilton or Lagrange formalismshown in subsection 3.3 for the double pendulum.

3.2 The harmonic oszillator

Another well known example of a conservative system is the harmonic oszillator described in section 2.3.

3.3 The double pedulum

A double pendulum is a system of two connected pendula. The first pendulum consists of a point massm1 and a cord of length l1 pendulum in subsection 3.2 and the second pendulum consists of a secondpoint mass m2 and a second cord of length l2 connecting the first and the second point mass. This systemhas two degrees of freedom, the displacements of the two point masses. The angle of elongation of thefirst point mass from the equilibrium we call θ1 and that of the second point mass θ2. Now we havefound the generalized coordinates and can use them to find the equations of motion with the HamiltonFormalism. We start with the kinetic energy:

T =m1

2(x2

1 + y21) +

m2

2(x2

2 + y22) (6)

where x1, y1 and x2 and y2 are the displacements of the first and second point mass in cartesian coordi-nates and x1, y1 and x2 and y2 are the velocities of both point masses.

In terms of generalized coordinates we get for the displacements the following equations:

x1 = l1 sin θ1

y1 = −l1 cos θ1

x2 = l1 sin θ1 + l2 sin θ2

y2 = −l1 cos θ1 − l2 cos θ2

and for the velocities:

x1 = l1θ1 cos θ1

y1 = l1θ1 sin θ1

x2 = l1θ1 cos θ1 + l2θ2 cos θ2

y2 = l1θ1 sin θ1 + l2θ2 sin θ2

7


When we use that equations we get the kinetic energy:

T =m1

2l21θ

21 +

m2

2(l21θ2

1 + l22θ22 + 2l1l2θ1θ2(cos θ1 cos θ2 + sin θ1 sin θ2)) (7)

The potential energy of the system is given by the equation:

V = m1gy1 + m2gy2 (8)

With the generalized coordinates we get:

V = −g((m1 + m2)l1 cos θ1 + m2l2cosθ2) (9)

With the potential and the kinetic energy we get the Lagrange’s function:

L =12(m1 + m2)l21θ2

1 +12m2l

22θ

22 + m2l1l2θ1θ2 cos (θ1 − θ2) + (m1 + m2)gl1 cos θ1 + m2gl2 cos θ2 (10)

Now we use the Euler-Lagrange’s equation

∂L

∂θi− d

dt

∂L

∂θi

= 0 (11)

to get the equations of motion

0 = gm2l2 sin θ2 + m2l22θ2 + m2l1l2θ1 cos(θ1 − θ2)−m2l1l2θ2

1sin(θ1 − θ2) (12)

0 = g(m1 + m2)l1 sin θ1 + (m1 + m2)l21θ1 + m2l1l2θ2 cos(θ1 − θ2) + m2l1l2θ22sin(θ1 − θ2) (13)

Now we can use this equations to calculate a numerical solution for the double pendulum.

Here we have to distinguish cases of small and larger angular diplacements. In the first case the experimentshows an almost pariodic behavior of the double pendulum and in the second case the behavior is chaotic.To simplify the solutions we set m1 = m2 = l1 = l2 = 1. The gravitational accelleration is g = 9.81.

3.3.1 Numerical solution for small elongations of the double pendulum

For the initial values θ1 = 1,θ2 = −1, θ1 = 0 and θ2 = 0 we get small oscillations (1(a) - 1(d)). Themotion is almost periodic like we observed it in the experiment.

8


5 10 15 20t

-1

-0.5

0.5

1

J1

(a) The deviation of the first mass θ1 is plot-ted in dependence on time

-1 -0.5 0.5 1J1

-4

-2

2

4

J 1

(b) Phase space: Displacement of the firstmass θ1 is plottet in dependence on the an-gular velocity of the first mass θ1

5 10 15 20t

-1

-0.5

0.5

1

J2

(c) Displacement of the first mass θ2 plottedin dependence on time

-1 -0.5 0.5 1J2

-6

-4

-2

2

4

6

J 2

(d) Phase space: Displacement of the firstmass θ2 plotted in dependence on the angularvelocity of the second mass θ2

Figure 1: Numerical solution for small elongations

3.3.2 Analytic solution for small elongations of the double pendulum

It is also possible to give an analytic solution concerning small angular displacements like it is done in[4]. The result shows that the masses should move in antiphase to each other, which is almost satisfiedin the illustrations of a numerical solution above.

3.3.3 Numerical solutions for larger elongations of the double pendulum

With initial values set θ1 = 1.5,θ2 = −1.5, θ1 = 0 and θ2 = 0 we examine larger elongations and chaoticmotion (2(a) - 2(d)).

These solutions show that the double pendulum is an example of a chaotic system.

9


10 20 30 40 50t

-2

-1

1

2J1

(a) Displacement of the first mass θ1 is plot-ted in dependence on time

-1.5 -1 -0.5 0.5 1 1.5J1

-6

-4

-2

2

4

6

J 1

(b) Phase space: Displacement of the first massθ1 is plotted in dependence on the angular ve-locity of the first mass θ1.

10 20 30 40 50t

-10

-5

5

J2

(c) Displacement on the first mass θ2 is plot-ted in dependence on time

-7.5 -5 -2.5 2.5 5 7.5J2

-10

-5

5

10J 2

(d) Phase space: Displacement of the first massθ2 is plotted in dependence on the angular ve-locity of the second mass θ2.

Figure 2: Numerical solutions for larger elongations

3.4 Conclusion

In nature we obtain occurence of periodic motion in different version. E.g. the motion of the eartharound the sun, the oscillation of atoms in molecules, biological cycles like the contraction and expansionof the human heart etc. Some of such processes can be simplificated and under preconditions described asconservative systems. In some cases it is additional possible to make an analytical analysis. The doublepedulum is a good example. In cases of small elongations we can find an analytical solution. For largerelongations we can only calculate a numerical solution and get chaos. Accordantly the numerical solutionis only an approximation and in most cases friction has an important affect on the motion of a system inorder that we can not describe it as a conservative systeme without making large faults. Some ways tosolve problems with consideration of friction will be discussed in the next section.

10

4 FORCED OSCILLATIONS

4 Forced oscillations

Now we investigate forced oscillations of a rotating pendulum. First we examine unforced dampedoscillations of a rotating pendulum (4.1) and afterwards undamped but forced ones (4.4). Finally weexamine an example for a chaotic oscillator: the damped and forced pendulum with an additional masschallenging the balance of the system.

4.1 Unforced damped oscillations

Hook’s Law describes undamped harmonic oscillations

T = −Dφ (14)

. Modelling damped oscillations an additional friction term is necessary

T = Jf = −Df − Cf (15)

0 = f + 2βf + ω2f (16)

. With J as the moment of inertia of the system, β = C2J as the damping factor, C as the friction

coefficient and ω2w = DJ , where D is the strength of the restoring force.

The system is described by a linear homogeneous differential equation of second order, which can besolved by an exponential function:

φ(t) = eb·t (17)

φ(t) = beb·t (18)

φ(t) = b2eb·t (19)

Combined with (16) we get the following characteristic equation.

0 = b2 + (2β)b + ω2 (20)

.

The solution of (20) isb = −β ± (β2 − ω2)

12 (21)

.

In the case, that β is smaller than ω, b is complex. When we plug the expression for b in our ansatz (23),and extract the real part from it with the euler equation we get:

f(t) = te−βt cos√

ω2 − β2t (22)

By comparing the relation of two successive amplitudes we get the damping factor or the logarithmicdecrement:

δ = ln (φ1

φ2) = βT (0) (23)

11


Because f(t) is supposed to be real there are two solutions. The first describes overdamped oscillationswhere β > w (23):

f(t) = e−β±√

β2−ω2(24)

β = w is refered to as the nonperiodic border-line case in which the oscillator does half a period and thenbehaves like in the overdamped case.

f(t) = e−βt (25)

4.2 Experiment 1: Angular frequency

We determined the angular eigenfrequency of the pendulum by measuring the period of oscillation re-peatedly and got the average < T >= (1, 844± 0, 030)s; so ω = 2 π

T = (3, 4± 0, 1)Hz

4.3 Experiment 2: Damping factor

The task was to measure the damping factor β using (23). We averaged 10 periods of large amplitudesand 20 periods of small amplitudes, so β is given by

β =ln φ1

φ11

10 · T0(26)

respectively

β =ln φ1

φ21

20 · T0(27)

.

In figure (3) we plotted β in dependence of the initial amplitude φ1. For small amplitudes the steep of βis very high and gets lower at higher amplitudes until it converges to a minimal value. The influence ofthe damping factor is stronger for small amplitudes, because energy of rotation depends on the square ofangular velocity ω while the damping factor only depends on ω.

0 2 4 6 8 10 12

Φ H°L

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

Β

Figure 3: Damping factor.

12


4.4 Forced damped oscillations

When a periodic external force affects the system equation (16) changes to

f + 2bf + ω2f = ω2g0 cos (ω′t) (28)

We have taken ω′ as the exciting frequency and g0 as the amplitude of the exciting oscillation. Thesolution can be found by a period ansatz

f(t) = f0 sin (ω′t− y) (29)

Using equation (28) and comparing the coefficients we get

ωg0 = (ω − ω′2)f0 cos (y) + 2βωf0 sin (Φ) (30)

ωg0 = −(ω − ω′2)f0 sin (y) + 2βωf0 sin (Φ) (31)(32)

with (31) phase shift is

tan (y) =2βω′

ω2 − ω′2(33)

Combining (30) and (33) we get an expression for the amplitude of the forced oscillation:

f0

g0=

ω2

ω2 − ω′2cos (y) (34)

=ω2

√(ω2 − ω′2)2 + (2βω′)2

(35)

By (29) we see that the system oscillates with a fixed frequency ω′ as long as a small damping affects it.The amplitude of the forced oscillation has a maximum for ωR =

√ω2 − 2β2.

This is effect is called resonance. A little damping causes an increasing amplitude of resonator’s oscillationwhich is called resonance-catastrophe.

4.5 Experiment 3: Amplification factor

The amplification factor follows from A = f0g0

≈ ω02b . So the damping factor becomes β = ω0

f0g0 =

(3, 41Hz)/(2 · 19) · 0, 4 = 0, 036. The value of β is higher than the one determined in the experiment 2.This is caused by a magnetic brake.

4.6 Experiment 4: Phase shift

WithΦ = arctan(2β

ω

ω2 − ω′2) (36)

the phase shift (Φ) was calculated from the values measured in experiment 3. In figure (4) you can seethat the phase shift between the exciting and the resulting oscillation, with a maximum of π

2 at ω = ω′.For frequencies that are far away from the eigenfrequency it decays to zero.

13


0.4 0.5 0.562 0.6 0.7

f HHzL

2

4

6

8

10

12

Φ H°L

0.4 0.5 0.562 0.6 0.7

Figure 4: Phaseshift against exiters frequency.

4.7 Experiment 5: Unforced damped harmonic oscillations

The amplitude of oscillation in appendix 7 stabilizes after a period of variation in the beginning. Theelliptic trjactories in phase space have almost the same half-axis. In appendix 5 the amplitude fallsexponentially as it is typical for damped oscillations. In phase space (appendix 6) the trajectories areelliptic while the half-axis becomes smaller in time.

4.8 Experiment 6: Chaotic oscillations

Placing a mass of unbalance in some distance from the center of the wheel the restoring forces becomenonlinear. These forces can be descibed by a potential which has two minima. The pendulum oscillatesunpredictable around the one or the other minima, the motion has become chaotic. The double minimapotential can also be seen in the phase space (appendix 10). Most of the time the pendulum moved inthe right potential minima and then changed to the other.

14

5 NUMERICAL ANALYSIS

5 Numerical analysis

The motion of Pohl’s rotating pendulum is governed by the equation of the damped and forced pendulum.We add a nonlinear term mgr sin(x) which represents a mass m placed at the distance r from the centerof the wheel.

Jx + δx + Dx + mgr sin x = F sin(ωt) (37)

Approximating the term sin(x) by a polynom of fourth order x − x3

6 and introducing the new scaleconstants α, ω0, β, ω we gain the equation of the Duffing oscillator:

x + αx + ω20x + βx3 = f sin(ωt) (38)

The new terms describe a nonlinear restoring force, which could be obtained by a double well springpotential like U(x) = k x2

2 + β x4

4 . So we got all neccessary conditions for chaos: the nonlinear potentialto reach sensitive dependence on the initial conditions, the damping term for loss of energy, the externalforce which supplies a third degree of freedom, so non intersecting chaotic trajectories are possible byuniqueness theorem. The uniqueness theorem states that two different trajectories cannot intersect inphase space. If they do, the solution for the initial value problem could not be unique.

15


-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

(a) T f = 0.33

100 110 120 130 140 150t

0.2

0.4

0.6

0.8

1

1.2

1.4

ΦHtL

(b)

-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

(c) 2T f = 0.35

100 110 120 130 140 150t

0.2

0.4

0.6

0.8

1

1.2

1.4

ΦHtL

(d)

-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

(e) 4T f = 0.3565

100 110 120 130 140 150t

0.2

0.4

0.6

0.8

1

1.2

1.4

ΦHtL

(f)

-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

(g) f = 0.4

0 20 40 60 80 100 120 140t

-1.5

-1

-0.5

0

0.5

1

1.5

ΦHtL

(h)

Figure 5: Period doubling route to chaos, shown without transients. Parameter are (α = 0.5, ω = 1,β = 1,ω2

0 = −1).

16


-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

(a) f = 0

0 5 10 15 20 25 30t

-1.2

-1

-0.8

-0.6

-0.4

-0.2

ΦHtL

(b) damped oscillations

-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

(c) f = 0.18

0 10 20 30 40 50t

-1.2

-1

-0.8

-0.6

-0.4

-0.2

ΦHtL

(d) attracted to the left potentialwell

-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

(e) f = 0.3

0 10 20 30 40 50 60 70t

0

0.2

0.4

0.6

0.8

1

1.2

ΦHtL

(f) attracted to the right poten-tial well

-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

(g) f = 0.4

0 20 40 60 80 100 120 140t

-1.5

-1

-0.5

0

0.5

1

1.5

ΦHtL

(h) chaotic oscillations

Figure 6: Discovering the Duffing equations phase space, initial conditions as seen in the pictures.

17


-1 -0.5 0 0.5 1 1.5ΦHtL

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Φ HtL

Figure 7: Poincare map of the strange attractor seen in figure 5(g). The chaotic attractor exhibits acharacteristic pattern by collecting only points in time steps of one period of the external force. Thisextracts order from chaos by considering a surface of section.

18

6 THE LORENZ SYSTEM

6 The Lorenz system

6.1 Intro

The Lorenz system is an autonomous set of three first-order nonlinear differential equations with threedegrees of freedom. The nonlinearities in this set are the quadratic terms xz and xy.

x = σ(y − x) (39)y = rx− y − xz (40)z = xy − bz (41)

The meteorologist Edward Lorenz deduced this equations as a simplified model for convection in theatmosphere. Furthermore it has some applications in laser dynamics and can be transformed to a modelfor a chaotic water wheel. In the Lorenz model we can observe damped and chaotic oscillations. Here wetry to point out some qualitative features about it.

6.2 Physical meaning

Figure 8: Convection cell

These equations describe a fluid system in the gravitational field heatedfrom below and cooled from the top, so that the difference of temperatureδT = Tw − Tc is hold constant. This is called a Rayleigh-Benard cell.

For small δT the fluid is at rest due to viscosity and heat is transported bymeans of thermal conduction. If δT is increased convection sets in, warmfluid elements move up, cool down and sink again. It comes to circulationin terms of regular convection rolls. If δT is further increased the systemvaries in time and the whole system becomes more complicated. Even theexternal force δT is independent of time the system exhibits a nontrivial timedependence.

The dimensionless system does not involve spatial variables. x(t) is proportional to velocity, so it mea-sures the amplitude of convection, while the system is limited to the phenomenon of convection rolls.y(t) measures horizontal temperature variations, while z(t) measures vertical ones. The parameter σestimates the ratio of kinematic viscosity to thermal conductivity; it is called Prandtl number. r is calledRayleigh number which is proportional to the temperature difference δT , while b is a geometric factor.All parameters are to be taken positive.

6.3 Linear stability analysis

Substituting x = y = z = 0 into (39-41), we got three equilibrium points:

X0 = (0, 0, 0) (42)

X± = (±√

b(r − 1),±√

b(r − 1), r − 1) (43)

.

19

6 THE LORENZ SYSTEM

0 0.2 0.4 0.6 0.8xHtL

0

0.2

0.4

0.6

0.8

1

yHtL

(a)

0 0.2 0.4 0.6 0.8xHtL

0

0.025

0.05

0.075

0.1

0.125

0.15

zHtL

(b)

0 0.2 0.4 0.6 0.8 1yHtL

0

0.025

0.05

0.075

0.1

0.125

0.15

zHtL

(c)

Figure 9: Lorenz system comes to rest for r = 0.5, σ = 10, b = 8/3.

0 2 4 6 8 10t

0.7

0.75

0.8

0.85

xHtL

(a) r = 1.3

0 1 2 3 4 5t

0

1

2

3

4

5

xHtL

yHtL

zHtL

(b) r = 5

-4

-2

0

2

4

xHtL

-5

-3

-1

01

3

5

yHtL

0

2

4

zHtL

-4

-2

0

2

4

xHtL

(c) Phase space for r = 5

Figure 10: Stationary state for 1 < r < rc, σ = 10, b = 8/3.

The system is clearly dissipative because ∇ · (x, y, z) = −(b + σ + 1) < 0.

By choosing r (∝ δT ) as the adjustable parameter, we fix the classical parameter set of σ = 10, whichcorresponds to the value for cold water and b = 8/3.

6.3.1 State of no convection

For r < 1 the origin is the only real equilibrium point corresponding to the stable non convective state.The trajectories are shown in figure (9). They are comparable to those of a strong damped pendulumwhen viscose damping is stronger than the eigenfrequency of the system.

6.3.2 Stationary convection rolls

For 1 < r < rc ≈ 24.74 the origin becomes unstable and two new equilibrium points arise; namely(x, y, z) = (±

√b(r − 1),±

√b(r − 1), r− 1). This is called a bifurcation point. Trajectories starting near

the origin tend to move away and will be attracted by one of the fix points. This does not mean thatthe fluid is at rest. Corresponding to the nonzero values of x(t), y(t), z(t) the fluid is circulating; onefix point stands for clockwise the other for counterclockwise circulation. For 1 < r < 1.346 the two newpoints turn out to be stable, while at 1.346 < r < rc they turn out to be stable spirals shown in figure(10).

20

6 THE LORENZ SYSTEM

6.3.3 Chaotic oscillations

The dynamics for r > rc become more complicated exhibiting chaotic behavior. We got three equilibriumpoints X0, X+ and X− and all of them are unstable. The trajectory in phase space (x-y-z) makes severalspirals around X− and then switches to X+ and so on. The two bands of the trajectory appear as wingsof a butterfly shown in figure (11).

-100

1020

xHtL

-20

0

20yHtL

0

20

40

zHtL

-100

1020

xHtL

-20

0

20yHtL

Figure 11: Lorenz attractor for r = 28, σ = 10, b = 8/3. Mention that the points X+ and X− havebecome the spiral centers of the butterfly-attractor.

-15 -10 -5 0 5 10 15 20xHtL

-20

-10

0

10

20

yHtL

(a)

-15 -10 -5 0 5 10 15 20xHtL

0

10

20

30

40

zHtL

(b)

-20 -10 0 10 20yHtL

0

10

20

30

40

zHtL

(c)

Figure 12: Lorenz attractor for r = 28, σ = 10, b = 8/3, X(0) = (0, 1, 0)

This behaviour is purely sensitive to the initial conditions. Even very small changes in the initial con-ditions lead to complete different trajectories. Which means that the number of spirals around X− andX+ is not predictable, if you do not know the exact initial conditions. So pertubation of the systemwill cause a loss of information about its time evolution though the system is fully deterministic. Interms of a definition given by Strogatz: Chaos is aperiodic long-term behaviour in a deterministic systemthat exhibits sensitive dependence on initial conditions.This means that two trajectories with slightlydifferent initial conditions can end up anywhere in the attractor. This sensitive dependence on the initialconditions is illustrated in figure (13); the difference in phase space of two trajectories is measured by

∆(t) =√

(x′(t)− x(t))2 + (y′(t)− y(t))2 + (z′(t)− z(t))2. (44)

Although the volume in phase space is contracting, nearby trajectories can end up anywhere in theattractor; this is what we call a strange attractor.

21

6 THE LORENZ SYSTEM

0 10 20 30 40 50 60 70t

0

10

20

30

40

50

DHtL

(a)

-15 -10 -5 0 5 10 15xHtL

10

20

30

40

zHtL

(b)

Figure 13: (13(a))Difference of two trajectories with slightly different initial conditions. X(0) = (0, 0.1, 0)& X ′(0) = (0, 0.1 + 1 · 10−9, 0). After a few oscillations the trajectories are completely different. (13(a))Trajectories in phase space in the range t = 35 . . . 45.

6.3.4 Coexistence of chaotic and nonchaotic behaviour

An interesting phenomenon is that for values of 24 < rc ≈ 24.74 both solutions, stable spirals and thestrange attractor are possible. This behaviour depends on the initial conditions. Areas in phase spaceleading to one of them are called basins of attraction even if they are difficult to calculate.

6.3.5 Period doubling route to chaos

Considering figure (14) we see periodic but nonharmonic oscillations for r ≈ 100.795.

Decreasing of r to 99.63 leads to oscillations of period 2T , further decreasing will lead to 4T , 8T periodsand so on. This accumulates at some critical value of r where completely aperiod chaotic oscillations ap-pear, which can be interpreted as infinite period oscilliations. This phenomenon is called period doubling,

22

6 THE LORENZ SYSTEM

-30 -20 -10 0 10 20xHtL

60

80

100

120

140

zHtL

(a) Period T orbit for r = 100.795

-30 -20 -10 0 10 20xHtL

60

80

100

120

140

zHtL

(b) Period 2T orbit for r = 99.63

-30 -20 -10 0 10 20 30xHtL

60

80

100

120

140

zHtL

(c) Chaotic Orbit for r = 99

Figure 14: Phenomenon of period doubling shown without transients.

which is concerned as a classical route to chaos occurring also in the Duffing oscillator experiment.

6.4

By variation of parameter r we have seen some qualitative features of the Lorenz system. As we have seenthe system can exhibit completly different kinds of motion. This implies the transition from the stateof equilibrium to a stationary state and its analogues to a damped pendulum. In addition to that wehave seen transition to chaos in the sense of a critical parameter set and in the sense of period doubling.We illustrated the characterics of a exemplary chaotic system with its sensitive dependence on the initialconditions. Furthermore we found coexistence of stationary and chaotic states related to the choice ofinitial conditions.

23

7 SELF-SUSTAINED SYSTEMS IN ELECTRICAL ENGINEERING

7 Self-sustained systems in electrical engineering

In this chapter we will discuss self-sustained oscillations used in the electrical engineering. Self sustainedoscillations are very important for electrical engineering. First of all the stability of amplitude and phaseeven if the oscillator was disturbed. So self-sustained systems are always used if there is a stable oscillationneeded. For example in radios, clocks, computers, micro-waves etc.

In the most cases of communication a self-sustained oscillation is used to produce a support frequency.This oscillation occures mostly in crystals or a circuit of a coil and a capacitor as frequency determingelements. This kinds of resonant circuits are not item of this discussion.

But it is also possible to create a self-sustained oscillator with capacitors and resistors as frequencydeterming elements. Two circuits of this kind are discussed in this section. The first is the astablemultivibrator. This is a simple circuit that can be discussed only with the knowledge of charging anddischarging a capacitor. The second is the relaxation oscillation which can also be discussed with basicknowledge of the capacitor. At the end of this section the behaviour of the capacitor in circuits withalternating current is discussed.

7.1 Charging and discharging a capacitor

It is possible to describe the charging and discharging by two simple homogeneous differential equations[3].For this DE expotential function is always a solution, so the functions that describe the behaviour ofcharging and discharging has the properties of expotential functions.

In the following we will show how to obtain these functions.

By Kirchhoff’s law the voltage over the capacitor (Uc) and over the resistor (Ur) is given by U0 = Ur +Uc.Using Ohm’s law and Q = C · U gives:

I =Ur

R=

U0 − Uc

R=

U0

R− Q

RC(45)

Simple differentiation of equation (45) and use of I = dQdt gives:

dI

dt=

1RC

· I

The function (46) is the solution of the DE.

I(t) = I0e− t

RC (46)

It would be usefull to get an equation for the voltage over the time. To get this equation we start withKirchhoff’s law:

Uc(t) = U0 − Ur (47)= U0 −R · I(t) (48)

= U0 −RI0e− t

RC (49)

= U0

(1− e−

tRC

)(50)

24


0.5 1 1.5 2 2.5 3t @sD

0.2

0.4

0.6

0.8

1U @VD I@AD

UHtL

IHtL

Figure 15: Current and voltage curves over time while charging a capacitor.

Plotting the equations (46) and (50) we get the curves shown in the figure (15).

To get the equation for discharging of a capacitor we must assume that the current I(t) = U(t)R is floating

through the capacitor. This current is the change of the charge in the capacitor. Therefore we get theDE (52). The solution of this DE is function (53) ploted in figure (16).

0.5 1 1.5 2 2.5 3t @sD

0.2

0.4

0.6

0.8

1U @VD

Figure 16: Voltage curve vs. time while discharging a capacitor.

I(t) =dQ

dt= − d

dt(CU) =

U(t)R

(51)

⇒ dU

dt= − U

RC(52)

⇒ U(t) = U0e− t

RC (53)

7.1.1 Technical applications

There are many technical applcations using a capacitor. For example a time-switch like it is shown infigure (17). In this circuit the capacitor will be charged slowly. While this time there will be a low voltageover the capacitor (see figure (15)) and the transistor will be switched off. If the capacitor is charged,the voltage over the capacitor will be high enough to switch on the transistor.

An other application is the short-time accumulator for a flashlight or a so called astable multivibrator1[5].

1This circuit produce a square-signal with a high number of harmonics. Because of the harmonics the circuit is calledastable multivibrator.

25


Figure 17: Scheme of a time-switch.

This oscillators are self-sustained system. They produce a stable output frequency with a constantamplitude. A astable multivibrator (see figure (18)) has two states. Following the state where transistorT1 is active will be called state 1 and the other state where transistor T2 is active will be called state 2.

Figure 18: Scheme of an astable multivibrator.

At the beginning, when the oscillator is connected to the voltage source, none of the transistors is active.But there is a high voltage at both bases of the transistors so that both have the tendency to switch on.By asymetrics of the circuit one transistor will switch first. For the following discussion we will assumethat the transistor T1 will switch on first. So the system gets into state 1.

If the transistor T1 switches on, the potential on the collector will be 0 and capacitor C1 will be chargedover resistor R2. The voltage at the basis of transistor T2 is determed by the voltage over capacitor C1.So this voltage is low until the capacitor is charged to a specific level.

In this time capacitor C2 will be discharged by the resistors R3 and R4. Then the capacitor will becharged by the base-emitter path of the transistor T1. At this time there is a voltage at the output that isnearly the voltage of the power supply. This state is stable until the voltage at the base of the transistorT2 reaches a level of ≈ 0.6V . In this moment transistor T2 will switch on and the circuit will reach state2.

In state 2 the voltage at the collector of the transistor will fall fast to 0 and also the output. CapacitorC2 will be discharged completely. This means that the voltage at the base of the transistor T1 will below and the transistor will switch off. While capacitor C2 is charging, the voltage at the base will riseand the capacitor C1 becomes discharged by resistors R1 and R2. This state is stable until the voltage

26


at the base of transistor T1 reaches the level of ≈ 0.6V and the circuit gets into state 1 again.

2 4 6 8 10t @sD

0.2

0.4

0.6

0.8

1U @VD

Figure 19: Signal of the astable multivibrator.

The time the circuit stays in the specific states determines on the time the capacitors need to be charged.So the period can be simply determined to be proportional to C1R2 + C2R1. If C1R2 = C2R1 the timethe circuit stay in state 1 is equal to the time the circuit will stay in the state 2 and a symmetric signalwill be at the output (figure (19)). If C1R2 6= C2R1 the circuit will stay in one state longer than into theother state.

7.2 Relaxation oscillation

Figure 20:

Another type of self-sustained oscillation are the relaxation oscillations (figure (20)). We examined theseoscillations by an experiment.

The circuit consists of a power supply (Ue), a capacitor C and a glow lamp (Gl). In the glow lamp, thereis an embedded resistor (Rgl) to reduce the current.

In the following we describe the relaxation oscillation and then obtain the equation for the relaxationoscillation.

The frequency determing parts of the circuit are the capacitor and the glow lamp. Because the behaviourof the capacitor is allready discussed in the sections above, we are only describing the functionality of theglow lamp[6]. A glow lamp consists of an anode and a cathode in a tube filled with gas. If the voltageover the electrodes is lower than the ignite-voltage, the free electrons in the gas do not get enough kineticenergy to ionise the gas. Raising the voltage between the electrodes, the kinetic energy of the electrones

27


will also raise until they reach the energy of ionisation. At this energy more electrons will be freed and aelectron-avalanche occurs and the glow lamp ignites. After the lamp ignited there are a lot of secondaryprocesses that participate at the gas-ignite. First of all the thermic heating-up of the gas is responsiblethat the minimal voltage for an glow-discharge is smaller than the ignite-voltage. If the gas is heated,the electrons do not need to have such a high kinetic energy for ionising processes.

This phenomenon is an important part of the relaxation oscillation. At the beginning the lamp does notglow and the capacitor is discharged. Now the circuit will be connected to the power-supply and capacitorwill be charged over the resistor R. While the capacitor is charged there is only a small voltage over theglow lamp. But this voltage will raise until the capacitor is charged to the level of the ignite-voltage. Ifthe glow-lamp will ignite, the resistor of the glow-lamp falls and the capacitor will be discharged. But atthe same time the voltage over the glow-lamp falls until it reaches the minimal voltage. Then the lampstop glowing and the capacitor will be charged again. This produces a stable (self-sustained) oscillationthat is powered by the voltage-supply.

For the mathematical discussion of the phenomenon it is helpfull to develop an equation for each stateof oscillation. The first state where the capacitor will be charged and the second where the glow-lampflashes.

For the first state we can say that the resistor of the glow-lamp is Rgl ≈ ∞. Therefore the capacitorwill be charged like it was discussed in the section (7.1) but with the modification that the capacitor wascharged to the level of the minimal voltage of the glow-lamp. So the voltage over the capacitor and overthe glow-lamp is given by the equation (54).

UGl(t) = Ue − (Ue − Ul)e−tR C (54)

td = RC · ln(

Ul − Ue

Uz − Ue

)(55)

The time the capacitor is charged until it reaches the ignite-voltage can be calculated by the equation(55).

Now we discuss the state where the glow-lamp flashes. In this state the glow-lamp have current-independent resistance Rgl and the capacitor will be discharged by this resistor until the voltage reachesthe minimum-voltage of the glow-lamp. Starting with the mesh-law 0 ≈ Ic + I, with I = U−U0

Rgland

Ic = CU we get the differential equation (56). The initial condition for the DE should be U(0) = Uz

because the voltage over the glow lamp just reached the value Uz. The solution for this DE is (57). Ifwe now set U(t) = Ul we get the time the glow lamp flashes (th).

0 = CU +U − U0

Rv(56)

U(t) = U0 + (Uz − U0)e−t

RglC (57)

th = RglC · ln(

Uz − U0

Ul − U0

)(58)

So the period of the oscillation is given by T = th + td.

The relaxation oscillations is a self-sustained oscillation. The oscillations are stable while the voltageof the power supply is greater than the ignition voltage of the glow-lamp. If the voltage of the power

28


supply goes against the ignite-voltage the period of the oscillation becomes ∞. This can be easily seenby equation (55).

In an experiment we have examined the relaxation-oscillation. At first we have measured the ignitionand minimal voltage of the glow lamp. For the ignition voltage we got a value of Uz = (147.52± 0.45)Vand for the minimal voltage we have got Ul = (127.2 ± 0.4)V . These values are needed to calculate theperiod. An other needed value is the the voltage U0 that can be calculated as the intersection of theregression of the I-U caracteristic and the y-axes. We’ve got a value of U0 = (122.56± 0.05)V .

In the following table all measured values and the calculated periods are listed.

Resistor (MΩ) Capacitor (µF ) Measured period (s) Calculated period (s) Aberration (%)1 1 0.55± 0.04 0.52 62.5 1 1.16± 0.02 1.24 60.5 2 0.66± 0.02 0.55 201 2 1.12± 0.04 1.03 90.5 4 2.16± 0.11 1.09 98

One can see that the most values match well but the last one must be a mistake while measurement.

7.3 Properties of alternating currents

For many technical applications the properties of the capacitor at alternating current is important. Firstof all the reactance [3] is a caracteristic value for the capacitor in circuits with alternating current.

To obtain equations for describing the properties of the capacitor we start with the equation U = QC . By

derivate after time we get the equation (59).

U =Q

C⇒ dU

dt=

d

dt

(Q

C

)=

I

C(59)

(a) Scheme

1 2 3 4 5 6t @sD

-1

-0.5

0.5

1U @VD I @AD

IHtL

UeHtL

(b)

Figure 21: Properties of the capacitor at alternating current circuits.

Assuming a time dependent input voltage Ue(t) we get equation (64). This equation describes the currentthat runs through the capacitor.

29


Ue = U0 cos (ωt) (60)dUe

dt= −U0ω sin (ωt) (61)

⇒ I = −U0Cω sin (ωt) (62)⇒ I = U0Cω cos (ωt + π) (63)⇒ I = I0 cos (ωt + π) with I0 = U0Cω (64)

Comparing the input voltage Ue (equation (60)) with the current we see, that the voltage hurry aheadthe current by π, independent from the input frequency. This is an important property of an capacitorin an alternating current circuit.

The imaginary resistance (impetance) of a capacitor Z can be found using the Ohm’s law with Z = UI .

This resistance (equation (66)) is a value depending on the frequency ω of the input voltage. Thisdependency is used by many technical applications. For example it can be used to filter specific frequenciesor whole parts of the frequency spectra.

|Z| = U0

I0=

1ωC

(65)

Z =1

iωC(66)

7.3.1 High- and low-pass filters

In an experiment we investigated the properties of the capacitor in circuits with alternating current calledhigh- and low-pass circuits. This circuits have the task to filter all signals having a higher (high-pass) ora lower (low-pass) frequency of an input signal.

(a) Low-pass circuit. (b) High-pass circuit.

Figure 22: Capacitors in typical filter applications.

Both circuits work like a voltage divider. But in contrast to the conventional voltage divider this onedepends on the frequency of the signal. Figure 22(a) shows the scheme of the circuit. One can see thevoltage divider. So we can use the equation (67) to calculate the output voltage U dependent on the

30


frequency ω and the input voltage Ue.

U =Z(ω)

R + Z(ω)· Ue (67)

=1

RiωC + 1· Ue (68)

⇒ |U | = 1√1 + ω2R2C2

· |Ue| (69)

But also there is a phase shift between the input and output voltage that depends on the frequency ofthe input voltage. This is given by φ = arctan(−ωRC). Analogous it is possible to obtain the equationfor the high-pass circuit (70) with a phase shift of φ = arctan

(1

ωRC

).

U =R

R + Z(ω)· Ue (70)

=R2ω2C2 − iRωC

1 + ω2R2C2· Ue (71)

⇒ |U | = ωRC√1 + ω2R2C2

· |Ue| (72)

2 4 6 8 10Ω @HzD

0.2

0.4

0.6

0.8

1

U @VD

High-pass

Low-pass

Figure 23: Output voltage over input frequency.

Using these results it is possible to simulate the behaviour of such a network with an input voltage weused in the experiment like the one given in (73).

Ue(t) =

1 0 < t < π

−1 π < t < 2π(73)

Ue(t) =4π

(sin (t) +

13

sin (3t) + · · ·)

(74)

To simulate the behaviour of the filters it is necessary to calculate the Fourier-series [2] of this function(74). Now we can use the functions (67) and (70) on each harmonic of the input signal and calculate theoutput signal by summing all new harmonics together with there the frequency dependent phase-shift.

31


2.5 5 7.5 10 12.5 15t @sD

-1.5

-1

-0.5

0.5

1

1.5U @VD

Figure 24: Input signal and the output signals of the high- and low-pass filters.

To simplify the calculation we have used only the first terms of 9th order. The result is plotted in thefigure (24).

.

32

8 VAN DER POL OSCILLATOR

8 Van der Pol Oscillator

8.1 Definition and discription

The Van der Pol Oscillator (figure 25) is a LCR oscillating circuit connected with a triode, power sourceand a coil with inductive resistance. The source supplies the circuit with energy and make the systembecomes a self-sustained one and the coil works as a nonlinear element by producing feedback.

Figure 25: The Van der Pol Oscillator

We can derive the differential equation that discribes the Van der Pol Oscillator like it is done in [1]. Theapplication of Kirchhoffs Law leads to the equation:

ULR = UC −→ LdiLdt

+ RiL =1C

∫icdt (75)

The current of the source ia is given by the equation:

ia = iL + iC (76)

where iL is the current of the coil and iC is the current of the capacitor.

Differentiation with respect to time of eq. (75) and the use of eq.(76) leads to:

Ld2iLdt2

+ RdiLdt

+iLC

=iaC

(77)

Differentiation of eq.(77) leads to

Ld3iLdt3

+ Rd2iLdt2

+1C

diLdt

=1C

diadt

(78)

The voltage at the grid of the triode can be written as:

ug = u0 + u where u = MdiLdt

with the mutial inductance M .

33


If we substitude this expressions in eq.(78)and rearrange it we get:

d2u

dt2+

R

L

du

dt+

u

LC=

M

LC

diadug

dug

dt(79)

The anode grid characteristic of the triode (figure 26) is approximately given by:

ia = i0 + S0u + S2u3 (80)

Figure 26: anode grid characteristic

The differentiation of this equation by ug and the substitution in eq.(79) leads to:

d2u

dt2− 1

LC(MS0 −RC + 3MS2u

2)du

dt+

1LC

u = 0 (81)

Then the Van der Pol equation is given by:

u− α(1− βu2)u + ω20u = 0 (82)

with α = (MS0 −RC)/LC, β = 3MS2/(RC −MS0), ω20 = 1/LC.

The Van der Pol equation can also be written in non-dimensional variables as:

x− a(1− x2)x + x = 0 (83)

with a = α/ω0, x = β12 u and the derivatives of x with respect to τ = ω0t.

By setting the derivatives of x to zero we see that the fixed point of the system is x = 0. Now we knowthat for the initial values x = 0 and x = 0 the system is stationary. But for several non-zero initial valueswe get some interesting numerical solutions and for different values of a the VdP oscillator behaves invery different ways.

8.2 Van der Pol oscillator as a harmonic oscillator

If we set a to zero the VdP equation reduces to

x + x = 0 (84)

34


5 10 15 20t

-2

-1

1

2

x

(a) x is plotted in dependence of the time. One cansee the systems harmonic behavior.

-3 -2 -1 1 2 3x

-3

-2

-1

1

2

3

x

(b) The phase space (x(x)) is plotted. Thetrajectories are circular.

red blue greenx 1 2 0x 0 0 1, 5

(c) three different sets of ini-tial values

Figure 27: VdPO as a harmonic oscillator

This equation describes a simple harmonic oscillator. With Mathematica we can find numerical solutions.

At figure (27(a)) one can obtain sinusoidally motion of the system.

35


8.3 Van der Pol oscillator as a self-sustained system

We get a very different behaviour of the VdPO if we set a = 1. Now the trajectories are attrected by thelimit cycle. The VdP equation becomes:

x− (1− x2)x + x = 0 (85)

In this case we get the nonlinear term x2 in the VdP equation and the system becomes self sustained.

5 10 15 20 25 30t

-2

-1

1

2

x

(a) x is plotted in dependence of time. One can seethat after some time the behavior of the system inthe two different cases of intial values become almostthe same.

-3 -2 -1 1 2 3x

-3

-2

-1

1

2

3

x

(b) Phase space (x(x)): The trajectories show clearlya attraction to a limit cycle.

red bluex −0, 001 −2, 5x 0 −2, 5

(c) two different sets of ini-tial values

Figure 28: VdPO as a selfsustained system

At figure (28(b)) one can see the limit cycle.

Several initial values lead after some time to a movement following that cycle.

This numerical result agrees with Lienard’s theorem. This theorem says that a system of the form

∂2x

∂t2+ f(x)

∂x

∂t+ g(x) = 0

has a limit cycle in the phase plane if five conditions are satisfied [7].

• f(x) and g(x) are continuously differentiable for all x

36


• g(x) is an odd function

• f(x) is an even function

• g(x) > 0 for all x > 0

• The odd function F (x) =∫ x

0f(u)du has exactly one positive zero at x = a, is negative for 0 < x < a,

is positive and never decreasing for x > a, and F (x) −→∞ for x −→∞

In our case f(x) = x2−1 is an even function and g(x) = x is an odd function with g(x) > 0 for all x > 0.Both functions are continously differentiable for all x. F (x) =

∫ x

0f(u)du =

∫ x

0(u2 − 1)du = x3

3 − x is anodd function (fig 8.3).

0.2 0.4 0.6 0.8 1x

-0.8

-0.6

-0.4

-0.2

0.2

f@xD

Figure 29: F (x)

a = 1 because F (1) = 13

3 − 1 = 0 and x(x2

3 − 1) < 0 for 0 < x < 1. F (x) is never decreasing for x < 1and F (x) −→∞ for x −→∞. Now we know that there should exist a limit cycle.

8.4 Relaxation oscillation

If we set a = 10 the Van der Pol oscillator shows relaxation oscillations. The VdP equation becomes:

x− 10(1− x2)x + x = 0 (86)

Also in this case the conditions for a limit cycle are satisfied.

We can see that the values of x are small when |x| decreases (relaxation) and growing fast near x = 0.When |x| increases x reaches a maximum (rapid jump) and decreases fast until |x| reaches its maximum.

37


10 20 30 40 50t

-2

-1

1

2

x

(a) x is plotted in dependence of the time. One canobtain the fast rise and the slow evolution.

-3 -2 -1 1 2 3x

-15

-10

-5

5

10

15

x

(b) The phase space (x(x)) is plotted.

red bluex 0, 0001 1x 0 −1

(c) two sets of initial val-ues

Figure 30: Relaxation oscillation

8.5 Van der Pol oscillator with hard excitation

If we approximate the anode-grid characteristic by a fifth degree polynomial eqn (80) becomes

ia = i0 + S0u + S2u3 + S4u

5 (87)

Now we get at eq.(81) an additional term 5MS4u4 and sustituting the expression θ = 5MS4/(RC−MS0)

eq.(82) becomes:u− α(1− βu2 − θu4)u + ω2

0u = 0 (88)

with a = α/ω0 and the derivativs of x with respect to τ = ω0t we get the equation:

x− a(1− βx2 − θx4)x + x = 0 (89)

For θ = −27, a = −0, 1 and β = 20 we get a hard excitation. The initial values of x and x should belarge to excite the system.

One can see that for different initial values we get two different kinds of behavior. For large values of xand x the system reacts like a self sustained system and we obtain the typical limit cycle, as representedby the blue and red trajectory. For smaller values, as shown by the green trajectory, the system behaveslike a dissipative system and convergates to the fixed point at the origin. This is called hard excitation.

Now we can examine the differential equation with Lienard’s theorem. f(x) = a(θx4 + βx2 − 1) andg(x) = x satisfy conditions of a limit circle. Integration of f(x) gives

F (x) = a(θx5

5+ β

x3

3− x) (90)

To find the solutions of F (x) = 0 we plot the function with θ = −27 and β = 20.

38


5 10 15 20t

-1

-0.5

0.5

1

x

(a) x is plotted in dependence on time.

-1.5 -1 -0.5 0.5 1 1.5x

-3

-2

-1

1

2

3

x

(b) Phase space (x(x))

red blue greenx −1 0, 338 0, 265x 2 0, 338 0, 265(c) two sets of initial values

Figure 31: VdPO with hard excitation

0.5 1 1.5 2 2.5 3x

-0.5

0.5

1

1.5

f@xD

Figure 32: F (x)

One can see that there are two solutions a and b. That result does not agree with Lienard’s theorem, butfor the solution b near x = 1 F (x) < 0 for a < x < b and F (x) is positive and not decreases for x > a.And for the a F (x) > 0. Now we can conclude that for several initial values of x the system behaves ina different way.

8.6 Synchronisation of the Van der Pol oscillator by an external force

If the Van der Pol oscillator is driven by an external force we get the equation:

x− a(1− x2)x + x = A sinΩt (91)

where A is the frequency and Ω is the amplitude of the external force. Once again we plot a numericalsolution.

We start with the initial values: a = −0, 1, ω = 1, Ω = 0, 2, A = 1, x = 0 and x = 1

Here the trajectory is attracted by the limit cycle.

We start with the initial values: a = −0, 1, ω = 1, Ω = 0, 2, A = 1, x = 0 and x = 0, 1

39


-1.5 -1 -0.5 0.5 1 1.5x

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

x

(a) The phase space (x(x)) of the system. One caneasily obtain the limit cycle that is reached aftersome time

50 100 150 200t

-1.5

-1

-0.5

0.5

1

1.5

x

(b) x is plotted in dependence on time.

Figure 33: Forced VdPO

50 100 150 200t

-1

-0.5

0.5

1

x

(a) x is plotted in dependence of the time.

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

ASinΩt

(b) x is plotted in dependence of the externalforce A sinΩt. The Lissajous figure has an an-gle of π

4, which indicates that the system and

the external force are in phase.

-1 -0.5 0.5 1x

-0.2-0.1

0.10.2

x

(c) The phase space (x(x)) of the system. Onecan see the limit cycle again.

Figure 34: Forced VdPO

If we compare the two different cases of initial values by obtaining the Lissajous figures for time intervalsof 35 seconds we see that in the second case with a smaller initial value of x the synchronisation is reachedfaster then in the first case.

40


-1 -0.5 0.5 1x

-1.5

-1

-0.5

0.5

1

1.5

ASinΩt

(a) t=0..35

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

ASinΩt

(b) t=35..70

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

ASinΩt

(c) t=70..105

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

ASinΩt

(d) t=105..135

Figure 35: Lissajous figures for x = 1

41


-1 -0.5 0.5 1x

-1

-0.5

0.5

1

ASinΩt

(a) t=0..35

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

ASinΩt

(b) t=35..70

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

ASinΩt

(c) t=70..105

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

ASinΩt

(d) t=105..135

Figure 36: Lissajous figures for x = 0, 1

42

9 OUTLOOK: SYNCHRONIZING CHAOS

9 Outlook: Synchronizing chaos

An outstanding phenomenon arises by coupling two of the Lorenz systems. Consider a set of equations:

x = σ(y − x) (92)y = rx− y − xz (93)z = xy − bz (94)

and second system coupled with it

u = x(t) (95)v = rx(t)− v − x(t)w (96)w = x(t)v − bw (97)

The second set is no more an autonomous one, since the first equation is no more a differential one.Mention that all nonlinearities are kept. Integrating the first set will give you the unpredictable chaoticsystem described above. The second set forced by the fist one will act on it in a surprising way. At firstsight there seems to be no correlation between them. But when transients are gone the sychronize theirtrajectories synchronize their motion in the chaotic attractor. This phenomenon of synchronization maybe give rise to a new way of data transfer and encryption...

43

REFERENCES

References

[1] Van der Pol Oscillator. 2006.

[2] Bronstein. Taschenbuch der Mathematik. Harri Deutsch.

[3] Wolfgang Demtroder. Experimentalphysik Band 2. Springer, 2004.

[4] Friedhelm Kuypers. Klassische Mechanik. Wiley-VCH, 2005.

[5] Wolfgang Lipps. Telekommunikation und Nachrichtelektronik in der Praxis. DARC Verlag, 1989.

[6] Thomas Pfeifer. Brockhaus abc Physik. F.A. Brockhaus Verlag, 1972.

[7] Steven H. Strogatz. Nonlinear Dynamics and Chaos. 1994.

44

oscillations - uni-potsdam.de · 2.3 simple harmonic oscillator ... had the idea that any...

Documents