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Algebraic orbits on period-3 window for the logistic map

Profile image of Antônio FidélisAntônio Fidélis
https://doi.org/10.1007/S11071-014-1719-0
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Abstract

Algebraic stable and unstable orbits are presented for the famous period-3 window of the logistic map x n+1 = r x n (1 − x n). It is exhibited the general polynomial that gives rise to both stable and unstable period-3 orbits. These orbits are shown for three different fixed control parameter values of r : at tangent bifurcation (birth), at super-stability and at ending pitchfork bifurcation (death) of the period-3 window. All orbits are exposed in two different ways: a sum of complex numbers x i = a + bc + bc, as proposed by Gordon (Math Mag 69:118-120, 1996), and via Euler's formula x i = a + 2|b| cos(θ). The algebraic expressions of a, b, c, |b| and θ are given for each r value for both stable and unstable orbits, as well as their numerical values and the Lyapunov exponent. It is shown that a and |b| are statistical quantities of the orbits.

FAQs

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What explains the significance of the period-3 window emergence in logistic maps?add

The period-3 window is the last periodic orbit to appear, indicating a transition to chaotic behavior, as shown by its critical role in Sharkovskii's ordering, which was established in 1963.

How are stable and unstable period-3 orbits identified mathematically?add

Both stable and unstable period-3 orbits arise from the roots of a polynomial of degree six, which is factored into stable and unstable components, with specific parameter values yielding their properties.

When does the period-3 orbit lose stability in the logistic map?add

The period-3 orbit loses stability at the control parameter value r ≈ 3, following a pitchfork bifurcation that generates the chaotic regime and leads to further bifurcations.

What method determines the algebraic roots for period-3 orbits?add

The method involves solving an eighth-degree polynomial, where roots are expressed using complex numbers in polar form, ensuring they yield real values consistent with the periodic orbits.

How do statistical properties of period-3 orbits vary with the control parameter?add

The mean value and standard deviation of x in period-3 orbits show contrasting trends: the mean decreases with r for stable orbits, while it increases for unstable orbits, reflecting the bifurcation dynamics.

Figures (6)
whose roots yield two period-3 orbits, such that one is the stable and the other is the unstable one. 2 The method
whose roots yield two period-3 orbits, such that one is the stable and the other is the unstable one. 2 The method
As said above, the point x = 1/2 belongs to the super-stable orbit; hence, this orbit can be written in a simpler way, starting from x9 = 1/2, as follows:
As said above, the point x = 1/2 belongs to the super-stable orbit; hence, this orbit can be written in a simpler way, starting from x9 = 1/2, as follows:
the orbit. Using the Gordon form, one gets Although there are only three points in each orbit, some statistical quantities can be obtained from the results above. For any of the orbits shown, regardless of r value or orbit stability, the mean value of x in this orbits is x = a and its standard deviation of the mean is Omean = |b| for the respective values of a and |b| of the orbit. Using the Gordon form, one gets above. For any of the orbits shown, regardless of r value 4 Discussion
the orbit. Using the Gordon form, one gets Although there are only three points in each orbit, some statistical quantities can be obtained from the results above. For any of the orbits shown, regardless of r value or orbit stability, the mean value of x in this orbits is x = a and its standard deviation of the mean is Omean = |b| for the respective values of a and |b| of the orbit. Using the Gordon form, one gets above. For any of the orbits shown, regardless of r value 4 Discussion
Table 1 continued
Table 1 continued
The numerical values of the algebraic quantities and the Lyapunov exponent are summarized in Table 1| for
The numerical values of the algebraic quantities and the Lyapunov exponent are summarized in Table 1| for
Fig. 1 Numerical evolution of a, Re(b), Im(b), |b| and 6/3 (upper part) and Lyapunov exponent along r interval of the period-3 window. The thicker lines and the subscript index “s” are related to the stable orbits, and the dotted lines and the sub- script index “wu” are related to the unstable ones. It is shown the angle 6/3, the actual value of the argument used in the cosine function in the first Euler’s formula, Eq. (5)
Fig. 1 Numerical evolution of a, Re(b), Im(b), |b| and 6/3 (upper part) and Lyapunov exponent along r interval of the period-3 window. The thicker lines and the subscript index “s” are related to the stable orbits, and the dotted lines and the sub- script index “wu” are related to the unstable ones. It is shown the angle 6/3, the actual value of the argument used in the cosine function in the first Euler’s formula, Eq. (5)

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References (7)

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