From Physics 111-Lab Wiki

I. Non-Linear Dynamics and Chaos

II. Pre-Labs must be printed separately. NLD Pre Lab Questions and Staff Sign Off Sheets.PRINT, FILL THIS OUT, Get it Signed by 111-Staff, turn in your Signed Pre-Lab Sheet with your report, to the 111-Lab Staff.

III. Experiment and Procedure

IV. The custom VI's for this experiment & Appendix

Before The Lab

Read some of the reprints to make sure you understand the theory behind the experiment. Review the pre-lab questions and look up the answers to those questions that you don't understand.

Discuss the Physics about this experiment with the faculty or the GSI's in the 111-Lab before starting.

You should keep a laboratory notebook. The notebook should contain a detailed record of everything that was done and how/why it was done, as well as all of the data and analysis, also with plenty of how/why entries. This will aid you when you write your report.

View the [NLD Video], discuss pre-lab questions with an instructor and get the Staff Sign-Off Sheet(NLD) signed.

Reprints and reading materials can be found on the Physics 111 Library Site

Prerequiste Reading Materials

S. H. Strogatz, Nonlinear Dynamics and Chaos, (Addison-Wesley, New York, 1994). \#Q172.5.C45 S767
This is an introductory book presuming no prior familiarity with the subject.
E. Ott, Chaos in Dynamical Systems, (Cambridge University Press, 1993). \#Q172.5.C45O87
An excellent, broad, introductory book for advanced students.
H. G. Schuster, Deterministic Chaos, an Introduction 2nd ed. (VCH New York, 1988). \#QC174.84.S381
An introductory book appropriate for those with an advanced mathematical background.
J. Theiler, "Estimating fractal dimension," J. Opt. Soc. Am. A 7, 1055 (1990).
Another review article with an excellent bibliography. This one focuses on fractals.
W. H. Press et al., Numerical Recipes in C (Oxford University Press, New York, 1988). \#QA76.73C15 N865
A compendium of numerical algorithms for all branches of science. It is included here for its discussion of the Fourier transform algorithm.
Abarbanel, "Making Physics From Chaos," videotape, \#QA845 A77, on reserve in the physics library.
An application of NLD to turbulence around torpedoes. The talk is clear but the speaker runs out of time.
R. Van Buskirk and C. D. Jeffries, "Observation of chaotic dynamics of coupled nonlinear oscillators," Phys. Rev. A 31, 3332 (1985). \#QC1.P412
A description of a driven pn junction resonator, similar to that of the laboratory experiment.
R. L. Zimmerman et al., "The electronic bouncing ball," Am. J. Phys. 60, 370 (1992). \#QC1.A4
B. K. Clark et al., "Fractal dimension of the strange attractor of the bouncing ball circuit," Am. J. Phys. 63, 157 (1995). \#QC1.A4
C. E. Shannon, The mathematical theory of communication, (University of Illinois Press, Urbana, 1964). \#TK5101 .S45
Kandoff, Leo P. "Chaos: A View of Complexity in the Physical Sciences", Encyclopedia Britannica(1986). (Excellent brief introduction.) pp. 63-92
Bai-Lin, Hao. "Chaos", World Scientific Publishers, Singapore, 1985. (General Introduction with reprints of papers in this field)In the Physics Library
Cvitanovic, P. "Universality in Chaos", Acta Physica Dolonica , A65 (No. 3), 1984, pp. 203-239 (1984)
May, Robert "Simple Mathematical Models with very Complicated Dynamics" Nature 261, pp.459-467 (1975). (Important early review of nonlinear maps.)
Feigenbaum, M. "Universal Behavior in Nonlinear Systems", Los Alamos Science, Summer 1980, pp. 4-27, (Introduction to universality; the quadratic map.)
"A Two-Dimensional Mapping with a Strange Attractor"Comm. Math. Phys. 50, pp. 69-77 (1976). (Introduction of the Hènon map; fractal strucure.)
Bracewell, R.N. "The Discrete Fourier Transform", ch.18-The Fourier Transform and its Applications, 2nd edition, McGraw Hill (1978). (Brief Introduction to the fast Fourier Transform.)
Testa, J., Perez, J., Jeffries, C. "Evidence for Universal Chaotic Behavior of a Driven Nonlinear Oscillator", Phys. Review Letters 48, pp. 714-717 (1982). (Experimental results on the driven pn junction; essentially the experimental system used in the Physics 111-Lab Experiment.)
Perez, J., Jeffries, C. "Effects of Aditive Noise on a Nonlinear Oscillator Exhibiting period doubling and Chaotic Behavior", Phys. Rev. B 26, pp. 3460-3462 (1982). (Experiment results on adding noise to the driven pn junction.)
Hamming, Richard W. "Maximum Entropy", The Art of Probability for Scientists and Engineers: Chp.7, pp. 253-265 + 7 pages of diagrams
Eckmann, J. P. "Ergodic Theory of Chaos and Strange Attractors", The American Physical Society: Vol. 57, No.3, July 1985. pp. 617- 656.
L. Wells and J. Travis LabVIEW for Everyone, Prentice Hall PTR (Available from Don Orlando).

Suggested Initial Reading: Ref. 1; Skim §1, §2.0-§2.2, §3.0-§3.4, §3.6, and then read §9.0, §9.2-9.4, §10.1-§10.5 (§10.6, §10.7 optional), §11, §12.0-§12.3

Introduction

This experiment is an introduction to Non-Linear Dynamics, Data Acquisition, Chaos theory and Fractals. Limited as we are by our senses and relatively short powers of recall, much of the physical world seems aperiodic and defies quantitative description. While we have yet to discover closed form solutions to the simplest of systems (e.g. the one-dimensional gravitational three-body problem), the field of chaos reveals structure in their dynamics. The results of chaos theory have found practical applications in almost every branch of science.

The term Chaotic has come to mean something very specific in this field. It does not mean random or unorganized as the ordinary English word does. It refers instead to a particular type of dynamical behavior. One which at first sight appears random, but underlying it are ordered deterministic laws. Chaotic systems exhibit these key features: They are aperiodic, they exhibit sensitive dependence on initial conditions, and they are in some sense bounded.

Theory

The PN Junction: an introductory Dynamical System

A dynamical system is essentially anything that varies with time. The pertinent variables necessary to describe its motion are called the dynamical variables and the evolution of the system describes a path, or trajectory in the state-space (also called phase space) of the dynamical variables. Dynamical systems may be either conservative: A volume element in state space remains invariant (recall Liouville's theorem; see Marion, J. B. Classical dynamics of particles & systems); or the system may be dissipative: for which regions of state space become compressed as the systems evolves.

One of the dynamical systems we study is a driven damped oscillator with a non-linear response. Because it is damped this is a dissipative system. It consists of a pn-junction (a diode) connected in series with an inductor L, a resistor R, and a driving sine wave oscillator of voltage $V_0(t)=V_{os}cos\left ({\omega }_0t \right )$ , see the left side of Figure 1.

Figure 1

Figure 2: Capacitance of a typical diode as a function of diode voltage. From Ref. 7.

The diode is the non-linear circuit element: Not only does it have an exponential I-V characteristic,

$I_d~(V_d)=I_0[exp(eV_d / kT)-1]$

but due to the properties of the junction \[see Ref. 8\], it has a non-linear capacitance as well. The capacitance may be written as a function of the voltage across the diode, V_d:

$C(V_d) = \begin{cases} C_0 exp(eV_d/kT):V_d >0 \\ \frac{C_0}{\sqrt {(1-eV_d/dT)}}:V_d\le 0 \end{cases}$ ,see Figure 2.

(Editor's Note 4/8/09: In the equation immediately above, the expression "dT" should be replaced by "kT". I'll fix this when the glitch with the equation editing feature of this wiki is repaired.)

Thus the diode may be modeled as a voltage-dependent capacitor connected in parallel with an ideal diode (a voltage-dependent current-source), giving the equivalent circuit on the right of Figure 1.

We use a computer to measure the voltage across the resistor R and the observed signal, V_s(t), is proportional to the current I(t).

For small values of driving voltage, the diode doesn't conduct, its capacitance is nearly constant, and the circuit behaves like a passive LRC resonator: A standard treatment of which yields a resonant frequency ${\omega }_{res}\approx\frac{1}{\sqrt{LC( V_d=0)}}=\frac{1}{LC_0}$ . But as the drive amplitude increases, the increasing diode capacitance drops the resonant frequency significantly.

The effect of increasing driving-amplitude is seen in the following figures. They are plots of I(t) (measured as V_s(t)in Volts) vs. time (measured in sample number of a 50 kHz clock). In all three plots the driving oscillator is operating at the same frequency, ω_res ≈ 3500 Hz (a period of ≈ 15 samples).


Figure 3: I(t). V_os ≈ 0.3 V, response period ≈ 15 samples.	Figure 4: I(t). V_os ≈ 0.5 V, response period ≈ 30 samples.	Figure 5: I(t). V_os ≈ 1.5 V, response period ≈ 60 samples

In Figure 3, V_os is set to a small value and the response signal V_s(t) a sine wave; this is the linear response. As V_os(t) is increased, V_s(t) begins to produce harmonic content at 2ω₀, 3ω₀ etc (not shown). But as you keep increasing the voltage, it will abruptly exhibit a change in behavior that, for reasons that will become clearer later, is called a bifurcation. This new state has subharmonic content ω₀/2 (Figure 4) and consequently is also called a period-doubling bifurcation. As V_os is further increased, this period doubling bifurcation occurs again and again at increasingly closely spaced values of V_os. As the period approaches infinity, the system becomes chaotic , i.e. it is aperiodic, with a small but visible noise-like signal component that looks like "jitter."

Figure 6: I(t). V_os ≈3.5 V,: The diode response current is now aperiodic.

As V_os is further increased, this "noise" dominates the trace; however, one finds narrow ranges of V_os ("windows") for which this noise completely disappears, leaving a periodic signal. This noise is not random noise due to fluctuation phenomena (e.g., Johnson noise, shot noise, flicker noise), but is a natural consequence of the non-linear equations of motion of the system. As such, it is determined by the solution of the equations. We call this noise chaos. When the system becomes chaotic, Fourier spectral analysis shows a transition from a discrete to a continuous distribution.

The above behavior, a period-doubling cascade to chaos, is observed in many physical, chemical, and biological systems. To better understand it we shift our attention from the time domain to state-space.

The equations (of motion) for the PN-junction (Ref. 8, Sec. II) are:

$\dot I = \frac{V_0-RI-V_d}{L}$ (Equation 1)

$\dot V_d = \frac{I-I_d(V_d)}{C(V_d)}$ (Equation 2)

$\dot \theta =\omega$ (the time derivative of the driving signal's phase) (Equation 3)

Notice that these equations do not depend explicitly on time (and so are called autonomous). Instead they relate the rate of change of the current state to the current state itself. That is, given a state-space M of the variables (I,V_d, $theta;), we may write equations 1-3 as one vector equation

$\dot{\vec p}=\vec F (\vec p)~where~p \in M$ (Equation 4)

Thus $\vec p$ denotes the current state of the system, and $\vec p(t)$ the path of the system through the three-dimensional state-space of the dynamical variables.

Phase-Path Reconstruction

If one could measure all the dynamical variables of a system simultaneously, the path $\vec p(t)$ could be plotted easily. However, in many cases this is not possible and moreover, it is, in most cases, not necessary. Look ahead to the reconstructed phase plots of Figures 7 to 10. These plots were made by sampling the diode current only. Rather than measuring the complete set of variables $(I,~V_d,~\theta)$ , the trajectory is available in each variable (i.e. we could have sampled the diode voltage and come up with similar plots).

Sampling only one dynamical variable and reconstructing the trajectory in a higher-dimensional space is a process called embedding: Given a discrete time series $\left \{I_n \right\}$ of regularly sampled values of $I_d~(t_0+n\tau)$ , where t₀ is an arbitrary initial time and τ is the sample period, we embed the current in a higher dimension (appropriately called the embedding-dimension, in this example, dimension 3) by forming the points $(I_n,~I_{n+1},~I_{n+2})$ for all data.

This set of points then reveals an object topologically equivalent to the actual path in state-space. That this works is rather surprising. Ott presents a precise explanation (Ref. 2, section 3.8), but the basic idea is that if you know the current at a given moment, I_n that information gives you a little bit of information about where you are in state space, that is, any point where the plane I=I_n intersects all possible trajectories. You then refine your knowledge by stipulating that you're at the point where not only is the current I_n, but the current was I_n-1 at a time τ in the past. This pins down the location further, and subsequent restrictions on what the current was in the past refines your knowledge about what state it's in now.

This is by no means obvious, and it's complicated by the fact that there do exist situations where it doesn't work. But that the geometric properties of the system's trajectory can be found by sampling only one of the state-variables is remarkable and useful: It means that the structure of large-scale, multi-dimensional dynamical systems is available to a simple, single-detector apparatus.

PN-Junction phase plots & The Bifurcation Route to Chaos

Figure 7- Figure 10 show reconstructed phase-space trajectories, $\vec p(t)$ , that correspond to the circuit's operation in Figure 3- Figure 6. The plots were made using the embedding technique described in the previous section, so the plot axes are $(I_n,~I_{n+1},~I_{n+2})$ . We speculate that θ revolves around the line z = y = x, and that the other two state variables, V and I, correspond loosely to the distance along that line and the distance from that line. The graphs are all stereo-optic. Try holding them at arm's length (or set them on a desk as you stand over them) and letting your eyes focus at infinity. Let two of the double images overlap so that you can see three images. The middle one, if you get it to match up just right, will look like it's 3D.

Figure 7: Left and Right stereo-optic images of the linear response of the PN-Junction. The path is almost an ellipse, which is the path of a simple harmonic oscillator.

For small values of the parameter V_os, the trajectory is a stable orbit. For larger value (after bifurcation to period 2) the orbit is still stable, but requires two cycles of the driving oscillator to return to its initial point (Figure 8). These trajectories are a subclass of a more general object called an attractor: The subset of state space toward which a dynamical system evolves. Because they are periodic (the paths are closed cycles), the simple attractors of Figure 7-Figure 9 are known as limit-cycles.

Figure 8: L & R images of the diode response after undergoing a period-doubling bifurcation

Figure 9: same as above after another bifurcation.

Increasing V_os to the point where the system is chaotic reveals a more complicated attractor: it is a fractal object and hence called a strange attractor.

Figure 10: The system is now chaotic. The path in phase space never closes. And the evolution traces out the system's attractor. To improve the visibility of the plot the lines connecting the points have been omitted.

Fractals

As you learned in calculus, most geometrical objects become smoother as one gets closer and closer to them. Fractals, however, remain twisty and crumpled no matter how close one scrutinizes them. As such, they defy conventional definitions of length and volume. Efforts to quantify fractals first led to several different definitions of dimension and finally to generalized dimension (Ref. 3, Section 5.3, Ref. 4 p1060, Ref. 2, p78), the most intuitive of which is:

bulk ∝ size^dimension, so that $dimension = \lim_{size \to 0}\frac{\mbox{ln} bulk}{\mbox{ln} size}$ for appropriate definitions of bulk and size. Note that taking the limit as size?0 means that this definition of dimension is a local property of the object and may vary over its extent.

For instance, in two dimensions a disk has an Area ∝ radius², and $\lim_{r \to 0}\frac{\mbox{ln}\pi r^2}{\mbox{ln}r}=2$ With this definition, objects of non-integral dimension are called fractals.

Flow and Liapunov Exponents

Chaotic systems are noted for several universal features: The bifurcation route to chaos; Stable windows of periodicity, the borders of which exhibit short bursts of chaotic signal ("intermittency"); And sensitivity to initial conditions. To help explain some of these features, we define a flow $φ t$ that maps a state $\vec p$ to its new state ${\phi}_t(\vec p)$ at a time t units later. This operation also satisfies the relationship:

$\phi_t(\phi_s(\vec p))=\phi_{t+s}(\vec p)$

Now consider two neighboring initial conditions $\vec P_0~\mbox{and}~\vec P_0+\vec h$ . Expanding $\phi_t(\vec P_0+\vec h)$ about $\vec P_0$ gives $\phi_t(\vec P_0+\vec h)= \phi_t(\vec P_0)+\bar{\bar J}(t)\vec h+O(\| h\|^2)+\ldots$ where $\bar{\bar J}(t)$ is the Jacobian matrix of partial derivatives of $p h i t$ with respect to $\vec P_0$ and $O(\| h\|^2)+\ldots$ are higher order terms. (In one dimension, $\bar{\bar J}(t)$ would just be $\frac{\partial\phi}{\partial x}(t)$ ). Thus we see that a small displacement in initial conditions $\vec h$ is magnified to $\bar{\bar J}(t)\vec h$ . The evolution of the distance between neighboring initial-conditions is then given by the eigenvalues of $\bar{\bar J}(t)$ . Liapunov (alternately spelled Lyapunov or Ljapunov) exponents are convenient measures of the eigenvalues and are defined as

$\lambda_n\equiv \lim_{t \to \infty}\frac{\mbox{ln}| n^{th}~eigenvalue~of~\bar {\bar J}| }{t}$ (Equation 5)

If the Liapunov exponents are all positive then the system diverges. If they are all negative then the system condenses to a point. If the largest one is positive and the sum of them all is negative, then it amounts to a stretch in one or more dimensions and a contraction in the others and chaotic behavior is guaranteed.

Recurrence relations

The equations of state (Eq's 1,2,3) are functions of a continuous time variable. Using the Runge-Kutta method of integrating ordinary differential equations, it is possible to form a set of discrete-time dynamical equations and these are seen to exhibit universal features common to other similar systems. For this reason, we study iterative maps: discrete versions of $φ t$ that relate the next state of a system to its previous state. You might wonder why, if chaotic systems are so sensitive to initial conditions, should we expect to see any useful results computed by an approximating method such as Runge-Kutta. The reason is explained in [Ref. 1, §10.1], where the authors present the 'shadowing' lemma: Although a finitely computed chaotic time-series, beginning with an initial value, will rapidly diverge from the true orbit, there exists some other orbit that coincides closely to the computed series.

Experiment and Procedure

The custom VI's for this experiement

Non-Linear Dynamics and Chaos

From Physics 111-Lab Wiki

Contents

Before The Lab

Prerequiste Reading Materials

Introduction

Theory

The PN Junction: an introductory Dynamical System

Phase-Path Reconstruction

PN-Junction phase plots & The Bifurcation Route to Chaos

Fractals

Flow and Liapunov Exponents

Recurrence relations

Experiment and Procedure

The custom VI's for this experiement

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