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(unpublished, 28 September 1986)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.


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 nth 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.