Bifurcation Diagrams Explorer

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Instructions

• Bifurcation diagrams are used in the study of dynamical systems and are applicable to a wide range of fields: from the modeling of biological populations, financial systems, and chemical reactions to the modeling of nonlinear circuits, spatial networks, and search algorithms, to mention a few (Citanović, 1989; Gulick, 1992; Moon, 1992; Garcia, 2018).

  Bifurcation Diagrams or Orbit Diagrams?

  • Bifurcation diagrams display collections of iterates (orbits) of a variable x as a parameter c is changed. Under iteration, a point can exhibit three types of behavior: attractive, repelling, or indifferent. When attractive and indifferent but no repelling points are displayed, a bifurcation diagram is often referred to as an orbit diagram. Some authors make a distinction between the two terms, calling bifurcation diagrams the collection of all periodic points (attracting, repelling, and indifferent). These collections form what they call P-curves. Both types of diagrams have a common feature: points accumulated resembling shadow curves called Q-curves (Ross, Odell, & Cremer, 2009; Ross & Sorensen, 2000; Neidinger & Annen, 1996). These P/Q-curves are well discussed in Ross' site. He has shown why some functions, like sin(c*x), lead to "broken" or incomplete orbit diagrams. Ross has also shown how to fix these.
  • At the time of writing, Google searches show that the "orbit vs. bifurcation" distinction is not common on the Web. In the nonlinear dynamics literature one can even find orbit diagrams been referred to as bifurcation diagrams. This and the fact that one can find collections of bifurcations in orbit diagrams might cause some confusion. We may resolve the resultant semantic ambiguities by considering that all orbit diagrams are bifurcation diagrams, but the reverse is not true. That is, an orbit diagram is a bifurcation diagram without repelling points.

• At this time our tool lets you explore many of the orbit diagrams found in the literature, providing a visualization of the underlying attractive behavior of a dynamical system. The tool implements many of the diagrams discussed in Robert L. Devaney books (1992; 1989; 1986) and by other authors. A world-class expert on Chaos and Dynamical Systems, Devaney is the Feld Family Professor of Teaching Excellence at Boston University (Wikipedia, 2018; Devaney, 1996). We have the honor of having a signed copy of the second edition of An Introduction to Chaotic Dynamical Systems (Devaney, 1986), on ocassion of an excellent talk Devaney delivered in the early '90s at ASU.
• Using the tool: To use the tool, complete all form fields; otherwise the tool will reset itself. Proceed as follows:
  • map: Select the map you are interested in from the selection menu. This will load default settings which can be accepted or modified as necessary.
  • range: Enter the desired range of values for parameter c and variable x. Ranges are subdivided in eight large increments.
  • color: Enter a color for the canvas, text font, graph, and pixels in short (e.g., c36) or long (e.g., cc3366) hexadecimal notation, only.
  • orbit: Enter the number of iterates n to use, the initial seed x₀, filter, and number of decimals places.

• First time users: If using the tool for the very first time, you may want to try the example provided. You may also want to accept the default settings.
• Zooming in/out: To zoom in/out a diagram, just edit the minimum and maximum values of c and x before submitting the form.
• View/Save: Use your browser to view/save the output. You can always copy/paste an image to the computer clipboard memory and crop it with Photoshop, Paint, or similar software. You may edit or use the generated images as you please and without copyright limitations.
• This tool is powered by our Minerazzi Grapher, a lightweight PHP class that generates all kind of graphs through a web browser. No additional libraries or software are needed.

What is computed?

• For a selected nonlinear map and each value of c, the program computes the orbit or sequence of n iterates of x.
• For instance for the quadratic map Qc = x² + c, n is usually set to 200 and a critical point is, though not always, used as the initial value or seed.

  A critical point is a point for which the first derivative of the function under examination vanishes; i.e., f'(x) = 0. Said critical point is degenerated if its second derivative also vanishes; i.e., f''(x) = 0. For the quadratic map, f(x) = Qc so f'(x) = 0 and f''(x) = 2. Therefore, x = 0 is a critical point for Qc, and it is nondegenerated. While it is true that seeds no need to be critical points, frequently the most interesting diagrams are obtained with a seed at or nearby a critical point.

• When the program is run, the computer calculates and plots the parameter c on the horizontal axis versus the asymptotic orbit of x on the vertical axis. That is, the program calculates, but does not plot, the first few iterations. This allows an orbit to settle down and avoids the displaying of transient effects so not plotting low iterates effectively acts as a filter. In this way we can grasp the true behavior of an orbit. For instance for the quadratic map, if n = 200 and we set the filter to 0.25, the program will not display the first 200*0.25 = 50 iterates, but only the last 200*(1 - 0.25) = 150. So if there is an attracting orbit for some value of c, that asympotic orbit is plotted.
• As browsers are known for crashing when processing a large amount of data, you may want to keep the number of iterates n relatively low, or accept a default value. How low or high you can go depends on the map to be explored. For most maps 100 < n < 300 with a filter of 0.25 is a fair setting. However for other maps, like the Henon map, n and the filter to be used can be set higher.
• To understand how transient effects impact a diagram, set the filter to 0. As expected, setting the filter to 1 will display nothing. A discussion of the effect of unfiltered low iterations is given in chapter 10 of Clifford Pickover's book (1990).
• For beautiful generations of fractals and some of the maps used by the tool, check Timothy Wegner and Mark Peterson's bundle (1991).

Who can use this tool?

• Students, teachers, and researchers interested in Chaos and Dynamical Systems.

Suggested Exercises

• Explore the Quadratic Map Qc = x² + c within the following c and x intervals.
  • For c: cmin = -1.525, cmax = -1.05; For x: xmin = -1.525, xmax = -1.05
  • For c: cmin = -1.425, cmax = -1.34; For x: xmin = -1.425, xmax = -1.34
  • For c: cmin = -1.406, cmax = -1.39; For x: xmin = -1.406, xmax = -1.39
  • For c: cmin = -1.8, cmax = -1.74; For x: xmin = -1.8, xmax = 1.5
  • For c: cmin = -1.788, cmax = -1.76; For x: xmin = -0.2, xmax = 0.2
  • For c: cmin = -1.65, cmax = -0.75; For x: xmin = -2, xmax = 2
  • For c: cmin = -1.8, cmax = -1.75; For x: xmin = -2, xmax = 2

• For the mentioned Quadratic Map
  • Describe the effects of using the following seed values: 0, 1, and 1.5.
  • Describe the effects of using the following n values: 100, 200, and 300.
  • Describe the effects of using the following filter values: 0, 0.25, and 0.70.

• Show that the diagram for the quadratic map Qc = x² - c (note the negative sign) is similar to that of the Logistic Map Lc = c*x*(1 - x).

References

• Cvitanović, P. (1989). Universality in Chaos. Second edition. Adam Hilger, New York.
• Devaney, R. L. (1992). A First Course in Chaotic Dynamical Systems: Theory and Experiment. Perseus Press, 1992. French Translation: Editions Addison-Wesley France, Paris. Japanese Translation, 1997, by Addison-Wesley.
• Devaney, R. L. (1989). Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics. Menlo Park, Calif.: Addison-Wesley, 1989. Italian Translation: Caos e Frattali---Matematica dei Sistemi Dinamici e Applicazioni al Calcolatore. Libreria Ulrico Hoepli, Milano, 1990. Dutch Translation: Chaos, Fractals, Dynamica: Computer-Experimenten in de Wiskunde. Addison-Wesley: Amsterdam.
• Devaney, R. L. (1986). An Introduction to Chaotic Dynamical Systems. Redwood City, Calif.: Addison-Wesley, 1986. Second Edition, 1989. Japanese Translation by Kyoritsu Press, 1988. Second Edition, 1990.
• Devaney, R. L. (1996). Brief Vita.
• Garcia, E. (2018). On PageRank, Spatial Networks, and Bifurcation Diagrams. IRThoughts Blog.
• Gulick, D. (1992). Encounters with Chaos. First edition. McGraw-Hill, New York.
• Moon, F. C. (1992). Chaotic and Fractal Dynamics. First edition. Wiley, New York.
• Neidinger, R. D. & Annen, R. J. (1996). The Road to Chaos is Filled with Polynomial Curves. The American Mathematical Monthly, Vol. 103, No. 8, p. 640-653.
• Pickover, C. A. (1990). Computers, Patterns, Chaos, and Beauty: Graphics from an unseen world. First edition. Chapter 10. St. Martin's Press, New York.
• Ross, C., Odell, M. & Cremer, S. (2009). The Shadow-Curves of the Orbit Diagram Permeate the Bifurcation Diagram, Too. International Journal of Bifurcation and Chaos, Vol. 19, No. 9, 3017-3031; World Scientific Publishing Company. See also Ross' site for additional useful articles and software.
• Ross, C. & Sorensen, J. (2000). Will the Real Bifurcation Diagram Please Stand Up!. The College Mathematics Journal, Volume 31, p. 2-14. Published Online: 30 Jan 2018. See also its doi.
• Wegner, T. & Peterson, M. (1991). Fractal Creations: Explore the Magic of Fractals on Your PC. Waite Group Press. Mill Valley, California.
• Wikipedia (2018). Robert L. Devaney.

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