Period-doubling bifurcation

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In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves.

A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system.

A period-doubling cascade is an infinite sequence of period-doubling bifurcations. Such cascades are a common route by which dynamical systems develop chaos.[1] In hydrodynamics, they are one of the possible routes to turbulence.[2]

Period-halving bifurcations (L) leading to order, followed by period-doubling bifurcations (R) leading to chaos.

Examples[edit]

Bifurcation diagram for the logistic map. It shows the attractor values, like and , as a function of the parameter .
Period-doubling cascade in an exponential mapping of the Mandelbrot set

Logistic map[edit]

The logistic map is

where is a function of the (discrete) time .[3] The parameter is assumed to lie in the interval , in which case is bounded on .

For between 1 and 3, converges to the stable fixed point . Then, for between 3 and 3.44949, converges to a permanent oscillation between two values and that depend on . As grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period doublings culminate at , beyond which more complex regimes appear. As increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near .

In the interval where the period is for some positive integer , not all the points actually have period . These are single points, rather than intervals. These points are said to be in unstable orbits, since nearby points do not approach the same orbit as them.

Kuramoto–Sivashinsky equation[edit]

Period doubling in the Kuramoto–Sivashinsky equation with periodic boundary conditions. The curves depict solutions of the Kuramoto–Sivashinsky equation projected onto the energy phase plane (E, dE/dt), where E is the L2-norm of the solution. For ν = 0.056, there exists a periodic orbit with period T ≈ 1.1759. Near ν ≈ 0.0558, this solution splits into 2 orbits, which further separate as ν is decreased. Exactly at the transitional value of ν, the new orbit (red-dashed) has double the period of the original. (However, as ν increases further, the ratio of periods deviates from exactly 2.)

The Kuramoto–Sivashinsky equation is an example of a spatiotemporally continuous dynamical system that exhibits period doubling. It is one of the most well-studied nonlinear partial differential equations, originally introduced as a model of flame front propagation.[4]

The one-dimensional Kuramoto–Sivashinsky equation is

A common choice for boundary conditions is spatial periodicity: .

For large values of , the solution evolves toward steady (time-dependent) solutions or simple periodic orbits. As is decreased, the dynamics eventually develops chaos. The transition from order to chaos occurs via a cascade of period-doubling bifurcations,[5][6] one of which is illustrated in the figure.

Logistic map for a modified Phillips curve[edit]

Consider the following logistical map for a modified Phillips curve:

where :

  • is the actual inflation
  • is the expected inflation,
  • u is the level of unemployment,
  • is the money supply growth rate.

Keeping and varying , the system undergoes period-doubling bifurcations and ultimately becomes chaotic.[citation needed]

Experimental observation[edit]

Period doubling has been observed in a number of experimental systems.[7] There is also experimental evidence of period-doubling cascades. For example, sequences of 4 period doublings have been observed in the dynamics of convection rolls in water and mercury.[8][9] Similarly, 4-5 doublings have been observed in certain nonlinear electronic circuits.[10][11][12] However, the experimental precision required to detect the ith doubling event in a cascade increases exponentially with i, making it difficult to observe more than 5 doubling events in a cascade.[13]

See also[edit]

Notes[edit]

  1. ^ Alligood (1996) et al., p. 532
  2. ^ Thorne, Kip S.; Blandford, Roger D. (2017). Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics. Princeton University Press. pp. 825–834. ISBN 9780691159027.
  3. ^ Strogatz (2015), pp. 360–373
  4. ^ Kalogirou, A.; Keaveny, E. E.; Papageorgiou, D. T. (2015). "An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 471 (2179): 20140932. doi:10.1098/rspa.2014.0932. ISSN 1364-5021.
  5. ^ Smyrlis, Y. S.; Papageorgiou, D. T. (1991). "Predicting chaos for infinite dimensional dynamical systems: the Kuramoto-Sivashinsky equation, a case study". Proceedings of the National Academy of Sciences. 88 (24): 11129–11132. doi:10.1073/pnas.88.24.11129. ISSN 0027-8424.
  6. ^ Papageorgiou, D.T.; Smyrlis, Y.S. (1991), "The route to chaos for the Kuramoto-Sivashinsky equation", Theoret. Comput. Fluid Dynamics, 3: 15–42, doi:10.1007/BF00271514, ISSN 1432-2250
  7. ^ see Strogatz (2015) for a review
  8. ^ Giglio, Marzio; Musazzi, Sergio; Perini, Umberto (1981). "Transition to Chaotic Behavior via a Reproducible Sequence of Period-Doubling Bifurcations". Physical Review Letters. 47 (4): 243–246. doi:10.1103/PhysRevLett.47.243. ISSN 0031-9007.
  9. ^ Libchaber, A.; Laroche, C.; Fauve, S. (1982). "Period doubling cascade in mercury, a quantitative measurement". Journal de Physique Lettres. 43 (7): 211–216. doi:10.1051/jphyslet:01982004307021100. ISSN 0302-072X.
  10. ^ Linsay, Paul S. (1981). "Period Doubling and Chaotic Behavior in a Driven Anharmonic Oscillator". Physical Review Letters. 47 (19): 1349–1352. doi:10.1103/PhysRevLett.47.1349. ISSN 0031-9007.
  11. ^ Testa, James; Pérez, José; Jeffries, Carson (1982). "Evidence for Universal Chaotic Behavior of a Driven Nonlinear Oscillator". Physical Review Letters. 48 (11): 714–717. doi:10.1103/PhysRevLett.48.714. ISSN 0031-9007.
  12. ^ Arecchi, F. T.; Lisi, F. (1982). "Hopping Mechanism Generating1fNoise in Nonlinear Systems". Physical Review Letters. 49 (2): 94–98. doi:10.1103/PhysRevLett.49.94. ISSN 0031-9007.
  13. ^ Strogatz (2015), pp. 360–373

References[edit]

  • Alligood, Kathleen T.; Sauer, Tim; Yorke, James (1996). Chaos: An Introduction to Dynamical Systems. Textbooks in Mathematical Sciences. Springer-Verlag New York. ISBN 978-0-387-94677-1. ISSN 1431-9381.
  • Giglio, Marzio; Musazzi, Sergio; Perini, Umberto (1981). "Transition to Chaotic Behavior via a Reproducible Sequence of Period-Doubling Bifurcations". Physical Review Letters. 47 (4): 243–246. doi:10.1103/PhysRevLett.47.243. ISSN 0031-9007.
  • Kalogirou, A.; Keaveny, E. E.; Papageorgiou, D. T. (2015). "An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 471 (2179): 20140932. doi:10.1098/rspa.2014.0932. ISSN 1364-5021.
  • Kuznetsov, Yuri A. (2004). Elements of Applied Bifurcation Theory. Applied Mathematical Sciences. 112 (3rd ed.). Springer-Verlag. ISBN 0-387-21906-4. Zbl 1082.37002.
  • Libchaber, A.; Laroche, C.; Fauve, S. (1982). "Period doubling cascade in mercury, a quantitative measurement". Journal de Physique Lettres. 43 (7): 211–216. doi:10.1051/jphyslet:01982004307021100. ISSN 0302-072X.
  • Papageorgiou, D.T.; Smyrlis, Y.S. (1991), "The route to chaos for the Kuramoto-Sivashinsky equation", Theoret. Comput. Fluid Dynamics, 3: 15–42, doi:10.1007/BF00271514, ISSN 1432-2250
  • Smyrlis, Y. S.; Papageorgiou, D. T. (1991). "Predicting chaos for infinite dimensional dynamical systems: the Kuramoto-Sivashinsky equation, a case study". Proceedings of the National Academy of Sciences. 88 (24): 11129–11132. doi:10.1073/pnas.88.24.11129. ISSN 0027-8424.
  • Strogatz, Steven (2015). Nonlinear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering (2nd ed.). CRC Press. ISBN 978-0813349107.
  • Cheung, P. Y.; Wong, A. Y. (1987). "Chaotic behavior and period doubling in plasmas". Physical Review Letters. 59 (5): 551–554. doi:10.1103/PhysRevLett.59.551. ISSN 0031-9007.

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