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We help you build a niche with perfect optimization that can elevate your rankings on Google search.

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We help you pin incredible ads that can assist you to generate propelling revenue.

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The brand value is the key aspect of every firm. Our sophisticated strategies help you reorganize the reputation management system.

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## Important Factors to Consider Before Choosing a Marketing Agency

With the growing need of a website for every business, it is also important to keep the website alive with good marketing strategies. Otherwise, other similar websites will take over the search engine rankings, which will cost the business a lot. Many new online businesses have closed down in the very first year due to poor marketing skills which did not help them enough to reach out to the maximum and right audience. Your business too needs a marketing team which can handle your online presence. While an in-house marketing team should be your first choice to build personalized strategies, hiring an agency can provide expert support to your marketing while easing your burden to build your brand.

It is important to understand how a marketing agency performs to be able to choose the right team for your business. Make sure that you keep these factors in check while hiring a marketing agency for your brand.

### Understand your goals

There are plenty of marketing strategies to grow any kind of business. But smart marketing only requires a few strategies that are meant specifically for the type of business it is being implemented upon. It is important to know your goals for at least a year before talking to a marketing agency. It will help you choose the right marketing plans which will deliver the best results without wasting time and money.

### Check for experience

A marketing agency with more experience will always be a better option as they have tested more marketing strategies with more brands than the newcomers in marketing. An experienced agency will always provide consultation and strategy session to make a marketing blueprint for your company.

### Ask for a portfolio

Mere marketing for multiple companies does not add to the experience. It also requires a record of successful campaigns. Review the agency’s portfolio to check the result and comments of its successful client endeavours. Try to connect with their client’s directly to get the best assurance for an agency’s promises.

### Check your budget

Marketing will require a great investment but will only help in generating more revenue. Know how big your budget is before looking for an agency. It will save your time on discussing your plans with only the agencies that are under your budget.

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### Check the channels

Know about the expertise of the marketing agency that you are planning to hire. Evaluate the channels in which the agency will be operating for marketing your business, such as website, blogs, SEO, social media. Understand the strengths and weakness of the agency in each channel to know whether they will be able to deliver to your requirements or not. A little bit of homework is necessary to reap better benefits in the future.

## How Technology Is Changing Dog Ownership?

Dogs are man’s best friend. If you believe this saying, then you should know that what affects a man will also affect his friend. The world is changing, and technology is a massive part of the changes happening.

Technology has not only affected man, though. It has also changed everything around him, including the ownership of his best friend, dogs.

The pet industry, in general, has experienced several technological advancements. There are new trends and innovations in pet care, grooming, and policies.

The benefits and challenges that came with dog ownership a few years ago are not the same with what dog owners today face. Technology has both improved and complicated certain things.

Are you a dog owner or planning to get one? We have highlighted the technological advantages that are currently available and improvements that you will see in a few years.

**What Has Technology Done To Dog Ownership?**

In 1915, Alpheus Hyatt Verrill said, “There is no excuse for pampering, constant fondling, dressing up in clothing and other ridiculous customs.”

Verrill, who insisted that dogs should be treated like the animals they are, will have more words than ‘ridiculous’ if he sees how dogs get treated today.

Americans pamper their dogs like children and probably spend more money on them than on their kids.

In 2018, Americans spent between $80 – $142 per month on pets, totaling about $1,300 per year. Despite economic and financial issues, pets, dogs especially, are still getting great treatments.

Dog owners understand the emotional benefits they get from their pets so, most of them do not mind the cost.

Technology has also helped to make the life of the dogs and their owners simpler and easier though more expensive.

Here are some of the aspects of dog owners have experienced changes;

**1. Gadgets**

Dog owners now have several devices to make monitoring and caring for their dogs easier. You can watch your dog in your absence, feed it treats, and even keep an eye on its health. You can also get a wide range of toys for your dogs.

One of the gadgets available for dog owners is the iCPooch, which allows you to video conference with your dogs when you are not home.

You can also remotely give them treats that will make the dogs come to the camera. PetBot also has a monitor that takes short videos of your pet and send them to your email address.

Apart from monitoring, you can also get gadgets to keep your dogs fit. A recent survey has shown that more than half of the American dog population is overweight.

You can keep your dog fit by getting a dog treadmill. Voyce, by Fitbit for dogs, will also help you record the activity levels of your dog, helping you to keep an eye on its health.

There is a wide range of gadgets available for dog owners today. Think of any way you can pet care can be more comfortable and you will find that there is a device already available for it. Even if there isn’t, the pet technology industry is regularly getting updated.

**2. Grooming**

You indeed love your dog, but you will agree that cleaning after your pet might be tiring at times. Technology has done a great deal in making the messy side of having a dog less messy.

From pet hair cleaning tools to grooming machines, you can enjoy your pets with less stress and hassle.

Groomrade, an innovation by Atronia Innovations, uses a noiseless air touch technology to help dog owners groom their dogs. Another advantage of this technology is that you get to avoid dog hair on your floor and carpet.

However, we don’t think technology will replace the services of an excellent grooming service soon, though.

Apart from what you do at home, you can still employ the services of a grooming center. You can get pet grooming in Minneapolis for a very reasonable price.

**3. Veterinary Services**

Technology has also improved veterinary services for dog owners. Dog owners can easily connect with their dog vets and a community of other dog owners. Gadgets designed to check the health of dogs also help the work of vets.

An example of these technological advancements is BabelBark. This platform, created by Roy Stein and Bill Rebozo, helps pet owners connect to a reliable network of local and affordable veterinarians. Technology has made vet services both cheaper and faster to obtain.

Dog owners can also connect to pet caregivers who can watch over their dogs. Mobile phones and security cameras ensure that dog owners can trust these caregivers to take the best care of their pets.

**4. Social Media**

It is no longer strange to see social media accounts open for various pets. Dog owners have not lagged in this trend. There are several social media accounts for dogs.

Facebook and Instagram are two of the most popular platforms for these accounts. You can open one for your dog today if you haven’t and join the network of dog lovers. You get to share awesome videos of your pet and any other information you feel like sharing.

**5. Research**

Technology has sponsored more profound inquiry into the life of pets. One of the most significant research is selective breeding. This research birthed the Great Danes and dachshunds.

Modern technology has taken it a step further though with select desired traits in species with precision in less time than traditional breeding takes.

Other research include the language of dogs, DNA editing, and much more. Research and innovations into pet wearable technology are also on the high rise with an expected 16% increase this year.

Technology has aided and pushed along research into dogs and pets in general, helping owners know more and get more from their pets.

**6. Robopets**

It would be wrong to talk about technology affecting dog owners without talking about robot pets.

With technological advancements and the issues surrounding pet sustainability, it wouldn’t be shocking if robot pets take the place of actual dogs in a few years.

AIBO, a robotic dog manufactured in Japan, was appreciated by owners. They even conducted funerals for the dog when Sony stopped repairs.

The busier lifestyle of pet owners and the expense of sustaining actual pets all point to a future for robotic pets.

However, in the meantime, according to a Fortune-Morning Consult Poll, 76% of pet owners see their pets as ‘beloved family members.’

While technology has helped improve pet grooming, care, and health, it is yet to replace man’s best friend.

## 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.

## 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.

## 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.

## 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).

xi

xii

A major goal of this project is to show how to integrate tools from these dierent

disciplines when studying nonlinear systems.

Many sections of this book develop one specic \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 condence 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 dierential

equations that introduces the qualitative theory of ordinary dierential 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 eects 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

xiii

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.

## 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)

## Nick’s Nonlinear Dynamics Archive

### Nicholas B. Tufillaro

This archive contains information and programs useful for nonlinear time series analysis with an emphasis on the new area of topological time series analysis and includes:

- Nonlinear dynamics resource letter from the September 1997 Issue of the American Journal of Physics . (PDF Version )
- Pointers to other nonlinear dynamics, time series analysis, and scientific software archives.
- References to some first papers in the topological analysis of chaotic time series,
- e-print servers at Los Alamos , and Stony Brook,
- Meetings, workshops, and conferences,
- Archives of software for the simulation and analysis of nonlinear systems,
- Movies of chaos and complexity,
- Name that nonlinear person, a picture gallery of people working in nonlinear science,
- Programs for the Macintosh computer showing nonlinear and chaotic dynamics (quadratic map, bouncing ball). And Ode, a program for the numerical solution of ordinary differential equations that runs on UNIX systems.

More information about Nick’s past (pre 1996) research and references to his work.

There is also a sample section of our book, An Experimental Approach to Nonlinear Dynamics and Chaos (Tufillaro/Abbott/Reilly), published by Addison-Wesley in 1992. [PDF]. [HTML Version]. Buy a complete copy at Amazon.

Nick’s esume showing some of the places he has visited and people he has worked with. Some physics courses Nick taught in Walla Walla which has plenty of water. Here is an audio introduction from Nick; a jig; or reel provides a more musical hello. Nick’s mug shot for crimes of fashion and picture gallery. A bike I built in 1996 after I lost my Bill Boston.

Pictures from my favorite place, Dunedin, New Zealand. Physics Department at University of Otago in Dunedin. The Otago Daily Times. My Dunedin diary [PDF] (circa 1993).

Previous Projects:

A lot can be learned about the dynamics of two-degree of freedom Hamiltonian systems by studying SAM–a Swinging Atwood’s Machine (circa 1982).

Bouncing ball (circa 1986).

Complex behavior in capillary ripples (shaken not stirred, circa 1988).

Our experiments on strings (like guitar strings) are reported here as well (circa 1990)

A talk outlining thoughts on the future of nonlinear dynamics (circa 1995).

Experiments on automated modeling of a chaotic pendulum and similar mechanical devices (circa 1996).

Talk overviewing potential applications of symbolic dynamics (circa 1997).

Examples of black-box (behavioral) modeling of electronic circuits.

Abstracts of our NSF sponsored research projects.

Also of Interest:

Argentine nonlinear dynamics group (Solari, Mindlin, Ponce-Dawson).

Web sites for Toby Hall and Andre deCarvalho studying horseshoe dynamics.

Predrag’s course on classical and quantum chaos.

UK Nonlinear News.

Marc Lefranc’s web site on The Topology of Chaos.

Phyiscs Jobs:

Chronicle of Higher Education current job listings in physics. AIP’s Academic jobs’s list. SIAM job list. LANL Jobs, Higher Education Jobs.

Music:

Summer folk concerts in King of Prussia, PA,

Bay Area Music Venues: New Music Bay Area, Old First, SF Conservatory of Music , Plough and Stars, Starry Plough Berkeley, Great America Music Hall, Freight and Salvage, The Fillmore, Cafe du Nord, Slims, Hotel Utah, House Concerts, Fox Theatre, Noe Valley Music Series

Directions to Deer Creek Site of Agilent Labs in Palo Alto:

Map to Agilent Deer Creek Labs in Palo Alto

Bay Area Transit Information

Directions to my San Francisco office.

## Bouncing Ball Program

The *bouncing ball simulation system *is a program for the Apple Macintosh computer that provides a physically accurate rendering of the motions of a ball impacting with a sinusoidally vibrating table. The program adheres to all the Macintosh interface guidelines, thus making simulations easy to run and examine. The program accompanies the text book *An Experimental Approach to Nonlinear Dynamics and Chaos *by Nicholas B. Tufillaro, Tyler Abbott, and Jerermiah Reilly (Addison-Wesley, 1992), and it is an excellent tool for learning about the behavior of chaotic systems. Chapter one of the book provides a good introduction to the basic periodic and chaotic dynamics of the bouncing ball system. The book also comes with an extensive User’s Manual. In addition to its pedagogical value, the system is also of great practical interest in several engineering applications, as well as in basic research.

The bouncing ball simulation system allows the user to see and hear (long experimental sounds file mp3) the periodic and chaotic impacts between the ball and table. The impacts can be mapped to various melodic and harmonic schemes, giving rise to a kind of chaotic music, or just noise. The motions can be viewed and analyzed by a multitude of methods. Here several windows on the ball’s motion are shown ranging from a simple plot of the trajectory of the ball and table (upper left), an animation of the ball’s motion (lower left), to the more abstract Poincare maps (right) clearly indicating the balls chaotic nature. Numerical data (upper right) can also be saved from a simulation for further analysis.

A closer examination of the strange attractor is possible by examining the *Poincare (Impact) Map* window in more detail:

or, a window on the *phase space *dynamics:

In addition to single simulations, the user can also investigate how the dynamics changes as the parameters are varied by constructing a *bifurcation diagram*.

The bifurcation diagram above clearly shows that the bouncing ball system can approach a chaotic motion by the now classic *period doubling route to chaos.* Indeed, as described in the book mentioned above, it is quite easy to set up an experimental apparatus (using a loud-speaker, a glass lens, a function generator, and a ball bearing) in which students can measure Feigenbaum’s delta for themselves.

Here are two pictures from the first bouncing ball experiment I set up in the basement of the Physics building at Bryn Mawr College (circa 1985).

Colorful and informative pictures are also provided by a *basin of attraction *diagram which shows the complex (fractal) dependence of the asymptotic solutions (multiple attractors) on the initial conditions.

A demonstration copy (or an updated version for registered users) of the program for evaluation by teachers who are considering using the text and software in the classroom is available by electronic transfer from the first author. Internet: nbt@reed.edu

The most recent research paper on the bouncing ball is here.

Here are some additional pictures of the experiment and experimental results showing the strange attactor using the electronics to construct a peak detector map:

## Period Doubling Ad Infinitum

This diagram reveals not one, but rather an infinite number of period doubling bifurcations. As is increased a period two orbit becomes a period four orbit, and this in turn becomes a period eight orbit, and so on. This sequence of period doubling bifurcations is known as a *period doubling cascade*. This process appears to converge at a finite value of around 3.57, beyond which a nonperiodic motion appears to exist. This period doubling cascade often occurs in nonlinear systems. For instance, a similar period doubling cascade occurs in the bouncing ball system (Figure 1.16). The period doubling route is one common way, but certainly not the only way, by which a nonlinear system can progress from a simple behavior (one or a few periodic orbits) to a complex behavior (chaotic motion and the existence of an infinity of unstable periodic orbits).

In 1976, Feigenbaum began to wonder about this period doubling cascade. He started playing some numerical games with the quadratic map using his HP65 hand-held calculator. His wondering soon led to a remarkable discovery. At the time, Feigenbaum knew that this period doubling cascade occurred in one-dimensional maps of the unit interval. He also had some evidence that it occurred in simple systems of nonlinear differential equations that model, for instance, the motion of a forced pendulum. In addition to looking at the qualitative similarities between these systems, he began to ask if there might be some *quantitative* similarity–that is, some numbers that might be the same in all these different systems exhibiting period doubling. If these numbers could be found, they would be “universal” in the sense that they would not depend on the specific details of the system.

Feigenbaum was inspired in his search, in part, by a very successful theory of universal numbers for second-order phase transitions in physics. A phase transition takes place in a system when a change of state occurs. During the 1970s it was discovered that there were quantitative measurements characterizing phase transitions that did not depend on the details of the substance used. Moreover, these universal numbers in the theory of phase transitions were successfully measured in countless experiments throughout the world. Feigenbaum wondered if there might be some similar universality theory for dissipative nonlinear systems [8].

By definition, such universal numbers are dimensionless; the specific mechanical details of the system must be scaled out of the problem. Feigenbaum began his search for universal numbers by examining the period-doubling cascade in the quadratic map. He recorded, with the help of his calculator, the values of at which the first few period-doubling bifurcations occur. We have listed the first eight values (orbits up to period ) in Table 2.1.

**Table 2.1:** Period doubling bifurcation values for the quadratic map.

While staring at this sequence of bifurcation points, Feigenbaum was immediately struck by the rapid convergence of this series. Indeed, he recognized that the convergence appears to follow that of a *geometric series*, similar to the one we saw in equation (1.35) when we studied the sticking solutions of the bouncing ball.

Let be the value of the *n*th period doubling bifurcation, and define as . Based on his inspiration, Feigenbaum guessed that this sequence obeys a geometric convergence,; that is,

where *c* is a constant, and is a constant greater than one. Using equation (2.26) and a little algebra it follows that if we define by

then is a dimensionless number characterizing the rate of convergence of the period doubling cascade.

The three constants in this discussion have been calculated as

The constant is now called “Feigenbaum’s delta,” because Feigenbaum went on to show that this number is universal in that it arises in a wide class of dissipative nonlinear systems that are close to the single-humped map. This number has been measured in experiments with chicken hearts, electronic circuits, lasers, chemical reactions, and liquids in their approach to a turbulent state, as well as the bouncing ball system [9].

To experimentally estimate Feigenbaum’s delta all one needs to do is measure the parameter values of the first few period doublings, and then substitute these numbers into equation (2.27). The geometric convergence of is a mixed blessing for the experimentalist. In practice it means that converges very rapidly to , so that only the first few ‘s are needed to get a good estimate of Feigenbaum’s delta. It also means that only the first few ‘s can be experimentally measured with any accuracy, since the higher ‘s bunch up too quickly to . To continue with more technical details of this story, see Rasband’s account of renormalization theory for the quadratic map [6].

Feigenbaum’s result is remarkable in two respects. Mathematically, he discovered a simple universal property occurring in a wide class of dynamical systems. Feigenbaum’s discovery is so simple and fundamental that it could have been made in 1930, or in 1830 for that matter. Still, he had some help from his calculator. It took a lot of numerical work to develop the intuition that led Feigenbaum to his discovery, and it seems unlikely that the computational work needed would have occurred without help from some sort of computational device such as a calculator or computer. Physically, Feigenbaum’s result is remarkable because it points the way toward a theory of nonlinear systems in which complicated differential equations, which even the fastest computers cannot solve, are replaced by simple models–such as the quadratic map–which capture the essence of a nonlinear problem, including its solution. The latter part of this story is still ongoing, and there are surely other gems to be discovered with some inspiration, perspiration, and maybe even a workstation.