1. Nonlinear Dynamics From the Harmonic Oscillator to the Fractals Invited Talk, Irkutsk, June 1999 Prof. Dr. R. Lincke Inst. für Experimentelle und Angewandte Physik der Universität Kiel The Physical Pendulum Modelling Using the EULER-Method STOKES- and Dry Friction Nonlinear Oscillation, Modelling and Experiments The POHL-Pendulum as Nonlinear Oscillator Sequence of Oscillations with Decreasing Damping POHL-Pendulum, Modelling according to EULER FEIGENBAUM-Diagram Sensitivity and Causality Regions and Fractals

2. The Harmonic Oscillator Approximated by a Physical Pendulum If one uses a potentiometer as transducer, the damped oscillation recorded does not show the expected exponential damping.

3. Damped Oscillations Modelling Using the EULER-Method A) Stokes‘ FrictionB) Dry Friction

4. Physical Pendulum With Low Friction Transducer: Hall Probe A) Stokes‘ FrictionB) Dry Friction

5. L = 7,34 mH = 787,1 Hz C = 5,57 µF T = 1,27 ms R L = 4,2   = R L /2L = 286/s R = 1 M  R L C (R L ) ADW Contrary to the mechanical oscillations, here the agreement between experiment and theory is excellent. LC Oscillations Measuring Frequency and Decay Constant

6. Nonlinear Oscillations Modelling with the EULER-Method If one adds a term in x 4 to the parabolic potential of a harmonic oscillator (i.e. a cubic term to the linear force), one ends up with a DUFFING-Potential. The resulting oscillation shows the typical behaviour: For nonlinear oszillators the period depends on the amplitude!

7. Nonlinear Mechanic Oscillations Some Other Experimental Setups Umlenkstift Decke Schraubenfeder Bahn A) Airtrack B) Physical Pendulum With Large Angles At 90° initial angle, the period is already 18% longer than at 5° F ~ x 3 C) Double Pendulum Period as Function of Initial Angle theory/experiment

8. Wackelschwinger A Nonlinear Oscillator Induction Coils Characteristics of a Nonlinear Oscillator: The time function is not harmonic. The period depends on the amplitude.

9. POHL Pendulum With Excentric Mass Paradigm for a Nonlinear Oscillator BMW Schrittmotor Wirbelstrombremse Unwucht  Position Transducer BMW The wellknown POHL Pendulum is a typical harmonic Oscillator used for demonstration purposes. With added excentric mass and a stepper motor it became a paradigm for the nonlinear oscillator. PC-Interface

10. POHL Pendulum Duffing-Potential V(  )=½k·  ²V d (  )=V(  )+D·(1+cos  )+C 4 0 3 2 1 By setting the eddy current damping accor- dingly, the following modes can be realized: 0: free oscillation, only mechanical friction 1: strong damping, quasi-harmonic 2: lesser damping, period doubling 3: lesser damping, period quadrupling 4: very low damping, chaos These four modes of oscillation are now investigated experimentally and with appropriate modelling. The added excentric mass transforms the parabolic potential of the harmonic POHL Pendulum into a DUFFING potential with ist characteristic two minima.

11. POHL Pendulum Position Transducer BMW Figures: The diagrams show the angle as function of time, the velocity as function of time and the velocity as function of angle (phase space). Left: No eddy current brake, only mechanical damping, no motor excitation. The pendulum was started in the extreme right position; it ends in the right potential minimum. Right: Excitation by stepping motor excenter, strong eddy current damping. The oscillation is nearly harmonic, the phase diagram is similar to an ellipse. 01

12. POHL Pendulum Position Transducer BMW Figures: In the diagrams angle as function of time and velocity as function of time, the motor phase is shown in addition. Left: The amplitudes of the oscillation alternate. One complete period of oscillation corresponds to two periods of the motor; this is called Period Doubling. Right: Four motor periods correspond to one complete oscillatory period. 23

13. POHL Pendulum Modelling with the EULER-Method Torque caused by added excentric mass. Equation of motion:  ·d 2  /dt 2 = - k·  -  ·d  /dt + A·cos(  ·t) -T·sin(  ) Abbreviations: Phi = , Phi1 = d  /dt, Phi2 = d 2  /dt 2 t := 0; Phi := 0; Phi1 := 0; REPEAT EULER-Method t := t + dt; Phi2 := - k/  ·Phi -  /  ·Phi1 + A/  · cos(  ·t) -mgR/  ·sin(  ); Phi1 := Phi1 + Phi2 ·dt; Phi := Phi + Phi1 ·dt; PutPixel (round(t),round(100-20·Phi),1); UNTIL t>639; In order to study further phenomena, we now continue with modellings of POHL Pendulumusing the algorithm shown here:

14. POINCARÉ Section An Alternative Representation For chaotic oscillations, neither the time function x(t) nor the phase diagram v(x) are very illuminating. Quite often deeper systematics appear, if one uses the phase diagram and marks those points that belong to a certain phase of the motor excitation (quasi ‘stroboscopically‘): such a representation is called Poincaré Section. The diagram on the left shows an oscillation with doubled period and thus two different Poincaré points. With period quadrupling, there would be four points and chaotic oscillations would show an infinite number. Often the Poincaré points assemble on marked trajectories, the socalled Strange Attractors.

15. POHL Pendulum Modelling with the EULER Method This program reproduces all measure- ments with the POHL Pendulum and the BMW transducer. Representative are the three aspects of a period quadrupling shown here. Time Function Phase Space4 Poincaré Points

16. POHL Pendulum Modelling with the EULER Method If the damping is decreased even further, the behaviour of the pendulum becomes chaotic: The time function shows no periodicity, and in the phase diagram the whole space is filled; the Poincaré points, however, gather in a ‘strange attractor‘ Phase Space Poincaré Points

17. POHL Pendulum Modelling with the EULER Method A systematic variation of the damping yields the socalled Feigenbaum Diagram. This plot here shows 100 extrema (max and min) after a transient period of 300 seconds. The marked positions correspond to the oscillatory modes shown before: 1. Simple oscillation. 2. Period doubling. 3. Period quadrupling. 4. Chaos. 5. Window in Chaos. 6. Free oscillation. 123 45 6

18. Sensitivity Towards Initial Conditions Causality and Deterministisc Chaos This picture shows the initial oscillations with starting angles of 50° and 50,2°. In the very beginning, no difference is visible. Close to the central maximum of the potential, the curves begin to separate, and soon they are completely different. The red trace gives the logarithm of their distance in phase space, i.e. log  (x 1 -x 2 )²+(v 1 -v 2 )² : the curves diverge exponentially. Weak Causality: Equal causes have equal effects. Strong Causality: Similar causes have similar effects. Deterministic Chaos: The sensitivity prevents long term predictions.

19. Sensitivity Towards Initial Conditions  o : 135.550°, 135.560°, 135.563° Even a very slight variation in the initial conditions can result in completely different oscillations. In the upper two pictures, the oscillations stabilize in the right half of the pendulum, in the left picture in the left half. A systematic investigation (varying the initial velocity also) produces two dimensional Domains for reaching stable oscillations in the right or left half of the POHL pendulum.

20. POHL Pendulum Domains for Settling in the Left/Right Half Repeated ZOOM: Self Similarity