Poincaré Map

 A continuous flow can generate a discrete map in at least two ways: by a time-T map and by a Poincaré map. A time-T map results when a flow is sampled at a fixed time interval T. That is, the flow is sampled whenever t = nT for n = 0, 1, 2, 3, and so on.

The more important way (as described, for instance, in Guckenheimer and Holmes [1]) in which a continuous flow generates a discrete map is via a Poincaré map. Let  be an orbit of a flow in  . As illustrated in Figure 4.2, it is often possible to find a local cross section  about  , which is of dimension n-1.

The cross section need not be planar; however, it must be transverse to the flow. All the orbits in the neighborhood of  must pass through  . The technical requirement is that for all  , where  is the unit normal vector to  at  . Let  be a point where  intersects  , and let  be a point in the neighborhood of  . Then the Poincaré map (or first return map) is defined by

 

where  is the time taken for an orbit starting at  to return to  . It is useful to define a Poincaré map in the neighborhood of a periodic orbit. If the orbit  is periodic of period T, then  . A periodic orbit that returns directly to itself is a fixed point of the Poincaré map. Moreover, an orbit starting at q close to p will have a return time close to T.

For forced systems, such as the Duffing oscillator studied in section 3.4, a global cross section and Poincaré map are easy to define since the phase space topology is  . All periodic orbits of a forced system have a period that is an integer multiple of the forcing period. In this situation, it is sensible to pick a planar global cross section that is transverse to  (see Fig. 3.8). The return time for this cross section is independent of position and equals the forcing period. In this special case, the Poincaré map is equivalent to a time-T map.

The Poincaré map for the example of the rotational flow generated by  is particularly trivial. A good cross section is defined by the positive half of the x-axis,  . All the orbits of the flow are fixed points in the cross section, so the Poincaré map is just the identity map, P(x) = x.

See Appendix E, Hénon’s Trick, for a discussion of the numerical calculation of a Poincaré map from a cross section.

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Linearization

 

 To calculate the stability of a fixed point consider a small perturbation, 

The Taylor expansion (substituting eq. (4.16) into eq. (4.2)) about  gives

It seems reasonable that the motion near the fixed point should be governed by the linear system

since  . If  is an equilibrium point, then  is a matrix with constant entries. We can immediately write down the solution to this linear system as

where  is the evolution operator for a linear system. If we let  denote the constant  matrix, then the linear evolution operator takes the form

where  denotes the  identity matrix.

The asymptotic stability of a fixed point can be determined by the eigenvalues of the linearized vector field  at  . In particular, we have the following test for asymptotic stability: an equilibrium solution of a nonlinear vector field is asymptotically stable if all the eigenvalues of the linearized vector field  have negative real parts.

If the real part of at least one eigenvalue exactly equals zero (and all the others are strictly less than zero) then the system is still linearly stable, but the original nonlinear system may or may not be stable.

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Magnetic Deflection of Electrons

If you have not already done so please read the General Introduction to Experiments EF-1 and EF-2.

In this experiment we consider the effect of a magnetic field on the motion of electrons. Just as the electric field represents the interaction of two charged particles at rest, the magnetic field represents the interaction of charged particles resulting from relative motion. The electric field can be defined as the force per unit charge, and the magnetic field in terms of the force on a unit current element.

A magnetic field can be produced by a current in a conductor. The field, in turn, exerts a force on a particle of charge q moving through it according to the Lorentz force equation

Thus an electron moving through a magnetic field is accelerated by a force with magnitude F proportional to the component of velocity perpendicular to the field, in a direction always perpendicular to both the field  and the instantaneous velocity . This relation between the directions of  and  has an immediate and important consequence: The magnetic field force never does work on the particle, since the particle always moves in a direction perpendicular to the force acting on it. For this reason such a particle moves with constant kinetic energy and thus with constant speed. The direction of the velocity, of course, can change and in this experiment you will observe the deflection of an electron beam by a magnetic field oriented perpendicular to the direction of the beam.

Consider the situation shown in the next figure. Electrons with charge  emerge from the electron gun with a speed v determined by the energy relation

just as in Experiment EF-1. Now the beam enters a region of length l in which there is a uniform magnetic field B (the source of which will be discussed later) oriented perpendicular to the plane of the figure, pointing out of the page. The resulting magnetic force has a magnitude F = evB and is always perpendicular to the velocity. Further, since the acceleration produced by this force is at each instant perpendicular to , only the direction of  changes.

Now the conditions just described are those needed to produce circular motion at constant speed. The radius R of the circular arc can be obtained from the equation for centripetal acceleration. Equating the centripetal and magnetic forces we have

After leaving the region of the magnetic field, the electrons again travel in a straight line, deflected by an angle  from the original axial direction. The length of the circular arc is S and hence . If  is small and R is large the arc length S will be approximately equal to l and the angle  will be small, giving

Also, from the figure, the transverse displacement a at the point of emergence from the field is

The beam strikes the screen at a point displaced a distance D from the undeflected beam position. The total displacement is given by

When the above expressions for  and a are substituted into this equation, the results are somewhat complicated. It can be simplified considerably by using the fact that  is small so that we can use the approximations  and . We then find the total deflection D to be

Using the energy relation, Eq. (1), we obtain

As this expression shows, the beam deflection is proportional to the magnetic field B. It depends inversely on the square root of the accelerating potential; in contrast to experiement EF-1 where the deflection varied as  itself, not its square root. The difference is that here we have an additional velocity dependence because of the nature of the magnetic force.

The magnetic field will be produced by a pair of solenoidal coils arranged as shown below. In the figure, the CRT face is shown end-on, while the solenoid coils are shown as if cut by a plane containing their longitudinal axis. Point C in the figure is at the center of the CRT screen and on the axis of the solenoids. The angles  and  are between lines from C to the ends of the coil windings and the axis. .5in

For a solenoid of infinite length the magnetic field is uniform along the axis and across the space inside the solenoid. The field strength is given by  where n is the number of turns per unit length of the coil, I is the current, and the constant . The expression for the field strength for one finite coil is given by

where the angles  and  are as indicated in the figure. They can be determined from the dimensions of the apparatus. Of course, the total field strength for two coils is twice the value calculated from Eqn. 3.

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Experimental Approach to Nonlinear Dynamics and Chaos

An Experimental Approach to Nonlinear Dynamics and Chaos is a textbook and a
reference work designed for advanced undergraduate and beginning graduate students. This book provides an elementary introduction to the basic theoretical and
experimental tools necessary to begin research into the nonlinear behavior of mechanical, electrical, optical, and other systems. A focus of the text is the description
of several desktop experiments, such as the nonlinear vibrations of a current-carrying
wire placed between the poles of an electromagnet and the chaotic patterns of a ball
bouncing on a vibrating table. Each of these experiments is ideally suited for the
small-scale environment of an undergraduate science laboratory.
In addition, the book includes software that simulates several systems described
in this text. The software provides the student with the opportunity to immediately explore nonlinear phenomena outside of the laboratory. The feedback of the
interactive computer simulations enhances the learning process by promoting the
formation and testing of experimental hypotheses. Taken together, the text and
associated software provide a hands-on introduction to recent theoretical and experimental discoveries in nonlinear dynamics.
Studies of nonlinear systems are truly interdisciplinary, ranging from experimental analyses of the rhythms of the human heart and brain to attempts at weather
prediction. Similarly, the tools needed to analyze nonlinear systems are also interdisciplinary and include techniques and methodologies from all the sciences. The
tools presented in the text include those of:
theoretical and applied mathematics (dynamical systems theory and perturbation theory),
theoretical physics (development of models for physical phenomena, application
of physical laws to explain the dynamics, and the topological characterization
of chaotic motions),
experimental physics (circuit diagrams and desktop experiments),
engineering (instabilities in mechanical, electrical, and optical systems), and
computer science (numerical algorithms in C and symbolic computations with
Mathematica).
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A major goal of this project is to show how to integrate tools from these di erent
disciplines when studying nonlinear systems.
Many sections of this book develop one speci c \tool” needed in the analysis
of a nonlinear system. Some of these tools are mathematical, such as the application of symbolic dynamics to nonlinear equations; some are experimental, such
as the necessary circuit elements required to construct an experimental surface of
section; and some are computational, such as the algorithms needed for calculating fractal dimensions from an experimental time series. We encourage students to
try out these tools on a system or experiment of their own design. To help with
this, Appendix I provides an overview of possible projects suitable for research by
an advanced undergraduate. Some of these projects are in acoustics (oscillations
in gas columns), hydrodynamics (convective loop|Lorenz equations, Hele-Shaw
cell, surface waves), mechanics (oscillations of beams, stability of bicycles, forced
pendulum, compass needle in oscillating B- eld, impact-oscillators, chaotic art mobiles, ball in a swinging track), optics (semiconductor laser instabilities, laser rate
equations), and other systems showing complex behavior in both space and time
(video-feedback, ferrohydrodynamics, capillary ripples).
This book can be used as a primary or reference text for both experimental
and theoretical courses. For instance, it can be used in a junior level mathematics
course that covers dynamical systems or as a reference or lab manual for junior
and senior level physics labs. In addition, it can serve as a reference manual for
demonstrations and, perhaps more importantly, as a source book for undergraduate
research projects. Finally, it could also be the basis for a new interdisciplinary
course in nonlinear dynamics. This new course would contain an equal mixture of
mathematics, physics, computing, and laboratory work. The primary goal of this
new course is to give students the desire, skills, and con dence needed to begin their
own research into nonlinear systems.
Regardless of her eld of study, a student pursuing the material in this book
should have a rm grounding in Newtonian physics and a course in di erential
equations that introduces the qualitative theory of ordinary di erential equations.
For the latter chapters, a good dose of mathematical maturity is also helpful.
To assist with this new course we are currently designing labs and software,
including complementary descriptions of the theory, for the bouncing ball system,
the double scroll LRC circuit, and a nonlinear string vibrations apparatus. The
bouncing ball package has been completed and consists of a mechanical apparatus
(a loudspeaker driven by a function generator and a ball bearing), the Bouncing Ball
simulation system for the Macintosh computer, and a lab manual. This package
has been used in the Bryn Mawr College Physics Laboratory since 1986.
This text is the rst step in our attempt to integrate nonlinear theory with easily
accessible experiments and software. It makes use of numerical algorithms, symbolic
packages, and simple experiments in showing how to attack and unravel nonlinear
problems. Because nonlinear e ects are commonly observed in everyday phenomena
(avalanches in sandpiles, a dripping faucet, frost on a window pane), they easily
capture the imagination and, more importantly, fall within the research capabilities
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of a young scientist. Many experiments in nonlinear dynamics are individual or
small group projects in which it is common for a student to follow an experiment
from conception to completion in an academic year.

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Quadratic Map Program

The Quadratic Map Program for the Apple Macintosh computer allows the user to easily run, analyze, see, and hear data from the quadratic map (aka, the logistic map):

which is the simplest example of a nonlinear system exhibiting chaotic and complex behavior. Like it’s cousin, the bouncing ball program, the quadratic map program strictly follows the Macintosh interface guidelines.

Several windows on the simulation can be exhibited at any given time:

As well as bifurcation diagrams :

showing how the system approaches chaos through the period doubling route to chaos.

By simply pointing an clicking, and capturing regions in bounding boxes, areas of interest can be expanded and explored in more detail:

The quadratic map program (and other software) are sold with the book: An Experimental Approach to Nonlinear Dynamics and Chaos by Nicholas B. Tufillaro, Tyler Abbott, and Jeremiah Reilly (Addison-Wesley, 1992). The book contains an extensive User’s Manual, as well as other instructional material providing a “hands on approach” to learning about nonlinear dynamics and chaos. For the lastest versions of the programs, or to get a copy for review or evaluation for possible class room use.

Nicholas B. Tufillaro,
Chaos Experiment Quadratic Map Lab Manual
(unpublished, 28 September 1986)

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