• Source: Period-doubling bifurcation

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. In hydrodynamics, they are one of the possible routes to turbulence.




Examples






= Logistic map

=
The logistic map is





x

n
+
1


=
r

x

n


(
1
−

x

n


)


{\displaystyle x_{n+1}=rx_{n}(1-x_{n})}


where




x

n




{\displaystyle x_{n}}

is a function of the (discrete) time



n
=
0
,
1
,
2
,
…


{\displaystyle n=0,1,2,\ldots }

. The parameter



r


{\displaystyle r}

is assumed to lie in the interval



[
0
,
4
]


{\displaystyle [0,4]}

, in which case




x

n




{\displaystyle x_{n}}

is bounded on



[
0
,
1
]


{\displaystyle [0,1]}

.
For



r


{\displaystyle r}

between 1 and 3,




x

n




{\displaystyle x_{n}}

converges to the stable fixed point




x

∗


=
(
r
−
1
)

/

r


{\displaystyle x_{*}=(r-1)/r}

. Then, for



r


{\displaystyle r}

between 3 and 3.44949,




x

n




{\displaystyle x_{n}}

converges to a permanent oscillation between two values




x

∗




{\displaystyle x_{*}}

and




x

∗

′



{\displaystyle x'_{*}}

that depend on



r


{\displaystyle r}

. As



r


{\displaystyle r}

grows larger, oscillations between 4 values, then 8, 16, 32, etc. appear. These period doublings culminate at



r
≈
3.56995


{\displaystyle r\approx 3.56995}

, beyond which more complex regimes appear. As



r


{\displaystyle r}

increases, there are some intervals where most starting values will converge to one or a small number of stable oscillations, such as near



r
=
3.83


{\displaystyle r=3.83}

.
In the interval where the period is




2

n




{\displaystyle 2^{n}}

for some positive integer



n


{\displaystyle n}

, not all the points actually have period




2

n




{\displaystyle 2^{n}}

. 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.




= Quadratic map

=
Real version of complex quadratic map is related with real slice of the Mandelbrot set.

















= Kuramoto–Sivashinsky equation

=

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.
The one-dimensional Kuramoto–Sivashinsky equation is





u

t


+
u

u

x


+

u

x
x


+
ν


u

x
x
x
x


=
0


{\displaystyle u_{t}+uu_{x}+u_{xx}+\nu \,u_{xxxx}=0}


A common choice for boundary conditions is spatial periodicity:



u
(
x
+
2
π
,
t
)
=
u
(
x
,
t
)


{\displaystyle u(x+2\pi ,t)=u(x,t)}

.
For large values of



ν


{\displaystyle \nu }

,



u
(
x
,
t
)


{\displaystyle u(x,t)}

evolves toward steady (time-independent) solutions or simple periodic orbits. As



ν


{\displaystyle \nu }

is decreased, the dynamics eventually develops chaos. The transition from order to chaos occurs via a cascade of period-doubling bifurcations, one of which is illustrated in the figure.




= Logistic map for a modified Phillips curve

=
Consider the following logistical map for a modified Phillips curve:





π

t


=
f
(

u

t


)
+
b

π

t


e




{\displaystyle \pi _{t}=f(u_{t})+b\pi _{t}^{e}}






π

t
+
1


=

π

t


e


+
c
(

π

t


−

π

t


e


)


{\displaystyle \pi _{t+1}=\pi _{t}^{e}+c(\pi _{t}-\pi _{t}^{e})}





f
(
u
)
=

β

1


+

β

2



e

−
u





{\displaystyle f(u)=\beta _{1}+\beta _{2}e^{-u}\,}





b
>
0
,
0
≤
c
≤
1
,



d
f


d
u



<
0


{\displaystyle b>0,0\leq c\leq 1,{\frac {df}{du}}<0}


where :




π


{\displaystyle \pi }

is the actual inflation





π

e




{\displaystyle \pi ^{e}}

is the expected inflation,
u is the level of unemployment,




m
−
π


{\displaystyle m-\pi }

is the money supply growth rate.
Keeping




β

1


=
−
2.5
,


β

2


=
20
,

c
=
0.75


{\displaystyle \beta _{1}=-2.5,\ \beta _{2}=20,\ c=0.75}

and varying



b


{\displaystyle b}

, the system undergoes period-doubling bifurcations and ultimately becomes chaotic.




Experimental observation


Period doubling has been observed in a number of experimental systems. 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. Similarly, 4-5 doublings have been observed in certain nonlinear electronic circuits. 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.




See also


List of chaotic maps
Complex quadratic map
Feigenbaum constants
Universality (dynamical systems)
Sharkovskii's theorem




Notes






References


Alligood, Kathleen T.; Sauer, Tim; Yorke, James (1996). Chaos: An Introduction to Dynamical Systems. Textbooks in Mathematical Sciences. Springer-Verlag New York. doi:10.1007/0-387-22492-0_3. 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. Bibcode:1981PhRvL..47..243G. 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. Bibcode:2015RSPSA.47140932K. doi:10.1098/rspa.2014.0932. ISSN 1364-5021. PMC 4528647. PMID 26345218.
Kuznetsov, Yuri A. (2004). Elements of Applied Bifurcation Theory. Applied Mathematical Sciences. Vol. 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" (PDF). 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 (1): 15–42, Bibcode:1991ThCFD...3...15P, doi:10.1007/BF00271514, hdl:2060/19910004329, ISSN 1432-2250, S2CID 116955014
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. Bibcode:1991PNAS...8811129S. doi:10.1073/pnas.88.24.11129. ISSN 0027-8424. PMC 53087. PMID 11607246.
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. Bibcode:1987PhRvL..59..551C. doi:10.1103/PhysRevLett.59.551. ISSN 0031-9007. PMID 10035803.




External links


Connecting period-doubling cascades to chaos

Kata Kunci Pencarian:

• Period-doubling bifurcation
• Bifurcation theory
• Bifurcation diagram
• Doubling
• Logistic map
• Feigenbaum constants
• Feigenbaum function
• Intermittency
• Attractor
• Mathematical constant