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Circle X Is Shown in the Diagram. Which Equation Can Be Used to Solve for Mã¢ë†â 1?

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Logistic Map

LogisticEquationIterations

Replacing the logistic equation

(dx)/(dt)=rx(1-x)

(i)

with the quadratic recurrence equation

x_(n+1)=rx_n(1-x_n),

(two)

where r (sometimes also denoted mu) is a positive constant sometimes known every bit the "biotic potential" gives the so-chosen logistic map. This quadratic map is capable of very complicated behavior. While John von Neumann had suggested using the logistic map x_(n+1)=4x_n(1-x_n) every bit a random number generator in the late 1940s, it was not until work by Westward. Ricker in 1954 and detailed analytic studies of logistic maps beginning in the 1950s with Paul Stein and Stanislaw Ulam that the complicated properties of this type of map beyond elementary oscillatory behavior were widely noted (Wolfram 2002, pp. 918-919).

The first few iterations of the logistic map (2) give

where x_0 is the initial value, plotted above through five iterations (with increasing iteration number indicated past colors; i is scarlet, ii is yellowish, iii is dark-green, 4 is blue, and 5 is violet) for various values of r.

The logistic map computed using a graphical procedure (Tabor 1989, p. 217) is known as a web diagram. A web diagram showing the first hundred or so iterations of this process r=3.741 and initial value x_0 approx 0.00079 appears on the cover of Packel (1996; left figure) and is animated in the correct figure higher up.

In general, this recurrence equation cannot be solved in closed form. Wolfram (2002, p. 1098) has postulated that any verbal solution must be of the form

x_n=1/2{1-f[r^nf^(-1)(1-2x_0)]},

(6)

where f is some part and f^(-1) is its inverse function. Thou. Trott (pers. comm.) has shown that smooth solutions cannot exist for generic values of r, with the possible exception of r fifty-fifty and nonzero. The only exact solutions known are for r=-two, r=2 and r=4, summarized in the tabular array below (Wolfram 2002, p. 1098), and R. Germundsson (pers. comm., Apr. 25, 2002) has proved that no other solutions of this form are possible.

LogisticEquationBifurcation

The illustration above shows a bifurcation diagram of the logistic map obtained by plotting as a part of r a serial of values for x_n obtained by starting with a random value x_0, iterating many times, and discarding the commencement points corresponding to values before the iterates converge to the attractor. In other words, the gear up of fixed points of x_n corresponding to a given value of r are plotted for values of r increasing to the right.

LogisticMapBifurcations

An enlargement of the previous diagram around r=3.5 is illustrated in a higher place, with value of r at which a 2^n-bike first appears indicated past blueish lines.

In society to report the fixed points of the logistic map, permit an initial point x_0 lie in the interval [0,1]. Now find appropriate weather on r which proceed points in the interval. The maximum value x_(n+1) tin can accept is found from

(dx_(n+1))/(dx_n)=r(1-2x_n)=0,

(7)

then the largest value of x_(n+1) occurs for x_n=1/2. Plugging this in, max(x_(n+1))=r/4. Therefore, to continue the map in the desired region, we must have r in [0,4]. The Jacobian is

J=|(dx_(n+1))/(dx_n)|=|r(1-2x_n)|,

(eight)

and the map is stable at a point x_0 if J(x_0)<1.

Now find the fixed points of the map, which occur when x_(n+1)=x_n. For convenience, drib the n subscript on x_n

f(x)=rx(1-x)=x

(9)

x[1-r(1-x)]=x(1-r+rx)=rx[x-(1-r^(-1))]=0,

(10)

then the stock-still points are x_1^((1))=0 and x_2^((1))=1-r^(-1).

An interesting matter happens if a value of r greater than iii is called. The map becomes unstable and we get a pitchfork bifurcation with two stable orbits of period two corresponding to the ii stable fixed points of f^2(x). The stock-still points of club two must satisfy x_(n+2)=x_n, so

For convenience, drop the n subscripts and rewrite

Notice that nosotros have establish the start-order fixed points besides, since two iterations of a beginning-guild fixed indicate produce a trivial 2d-order fixed point. The truthful 2-cycles are given past solutions to the quadratic part

These solutions are simply real for r>=3, and then this is where the 2-wheel begins. Note that the 2-cycle can also exist plant past computing the discriminant of

(f^2(x)-x)/(f(x)-x)=r^2x^2-r(1+r)x+(1+r)=0,

(22)

which is

((1+r)(3-r))/(r^2).

(23)

When this equals 0, two roots coincide, so r_2=3 is the onset of period doubling. For n=2, the solutions (x_1,x_2,r) are given by (0, 0, +/-1) and (2/3, 2/3, 3), so the beginning bifurcation occurs at r_2=3.

In full general, the set of n+1 equations which tin be solved to requite the onset of an arbitrary n-wheel (Saha and Strogatz 1995) is

{x_2=rx_1(1-x_1); x_3=rx_2(1-x_2); |; x_n=rx_(n-1)(1-x_(n-1)); x_1=rx_n(1-x_n); r^nproduct_(k=1)^(n)(1-2x_k)=1.

(24)

The first n of these give f(x), f^2(x), ..., f^n(x), and the last uses the fact that the onset of period n occurs past a fold bifurcation, and so the nth derivative is i. For pocket-size n, these tin be solved exactly, but the complexity rapidly increases with n.

Now look for the onset of the three-bicycle. To eliminate the ane-cycles, consider

(f^3(x)-x)/(f(x)-x)=0.

(25)

This gives

1+r+r^2-(r^4+2r^3+2r^2+r)x+(2r^5+3r^4+3r^3+r^2)x^2-(r^6+5r^5+3r^4+r^3)x^3+(3r^6+4r^5+r^4)x^4-(3r^6-r^5)x^5+r^6x^6=0.

(26)

The roots of this equation are all imaginary for r less than some cutoff r_3, at which point two of them convert to real roots. The value of r_3 can be institute by computing the discriminant of (26),

D=((r^2-5r+7)^2(r^2-2r-7)^3(1+r+r^2)^2)/(r^(30)).

(27)

When the discriminant is zip, two roots coincide. This happens at

r_3=1+2sqrt(2)=3.828427...

(28)

(OEIS A086178) as first shown by Myrberg in 1958, so the 3-wheel starts at r_3. Saha and Strogatz (1995) give a simplified algebraic treatment for the 3-wheel which involves solving

r^3(1-2alpha+4beta-8gamma)=1,

(29)

together with 3 other simultaneous equations, where

Further simplifications still are provided in Bechhoeffer (1996) and Gordon (1996), simply neither of these techniques generalizes easily to higher cycles. Bechhoeffer (1996) expresses the three boosted equations as

giving

r^2-2r-7=0.

(36)

This has the positive solution found previously, r_3=1+2sqrt(2).

Gordon (1996) derives not only the value for the onset of the 3-cycle, just also an upper jump r^' for the r-values supporting stable catamenia-iii orbits. This value is related to the unique positive root s_1 of the cubic equation

s^3-11s^2+37s-108=0

(37)

by

r^'=1+sqrt(s_1),

(38)

which is the unique positive root of the sextic polynomial

(OEIS A086179). For n=3,

Solving the resulting cubic equation using estimator algebra gives

r=1+2sqrt(2)

(44)

and x_1, x_2, x_3 given by

343x^6-980x^5+868x^4-134x^3-161x^2+70x-7=0,

(45)

giving numerical roots

where 2+2cos(2/7pi) is the silvery constant.

To observe the onset of the 4-cycle, eliminate the 2- and i-cycles by considering

(f^4(x)-x)/(f^2(x)-x)=0.

(56)

This gives a 12th-order polynomial in x. The value of r_4 can be found past computing the discriminant of this polynomial,

D=((r^2+1)^3(r^2-4r+5)^3(r^2-2r-5)^2)/(r^(132))(r^6-6r^5+3r^4+28r^3-9r^2-54r-135),

(57)

whose only real positive roots are

The 4-cycle therefore starts at

r_4=1+sqrt(6)=3.449489...

(62)

(OEIS A086180).

The onset of 5-cycles tin be institute analytically, and gives a 22nd-social club polynomial in r whose real positive roots are r_5=3.73817... (OEIS A118452), 3.90557..., and 3.99026....

The onset of 6-cycles can be found analytically, and gives a 40th-order polynomial in r whose real positive roots are r_6=3.62655... (OEIS A118453), 3.93751..., 3.97776..., and 3.99758....

The onset of seven-cycles tin be constitute analytically, and gives a 114th-order polynomial in r whose existent positive roots are r_7=3.70164... (OEIS A118746), 3.77413..., 3.88602..., 3.92218..., 3.95102..., 3.96897..., 3.98474..., 3.99453..., and 3.99939....

The onset of viii-cycles can be found analytically as the polynomial root of the 8th-order polynomial

(OEIS A086181; Bailey 1993; Bailey and Broadhurst 2000; Borwein and Bailey 2003, pp. 51-52).

The onset of 16-cycles at r_(16)=3.564407266095... (OEIS A091517) was originally found using an integer relation adding which determined that alpha=r_(16)(r_(16)-2) is the root of a 120th-degree integer polynomial with coefficients that decrease monotonically from 257^(30) to ane (Bailey and Broadhurst 2000; Borwein and Bailey 2003, pp. 52-53). This consequence was afterwards verified exactly using computer algebra (Borwein and Bailey 2003, p. 53; Kotsireas and Karamanos 2004), and is an algebraic number of degree 240.

The following table summarizes the values r_n at which the n-cycle start appears. For n=1, 2, ..., these have algebraic degrees 1, ane, 2, 2, 22, xl, 114, 12, ... (OEIS A118454).

The algebraic orders of the values of r_(2^n) (i.e., the onset of the 2^n-wheel) for n=1, two, ... are therefore given past one, 2, 12, 240, ... (OEIS A087046). A table of the bike type and value of r_(2^n) at which the cycle 2^n appears is given below.

For boosted values, run into Rasband (1990, p. 23). Note that the table in Tabor (1989, p. 222) is wrong, as is the n=2 entry in Lauwerier (1991). The menstruum doubling bifurcations come faster and faster (8, 16, 32, ...), then suddenly break off. Beyond a certain bespeak known as the aggregating point, periodicity gives way to chaos, as illustrated beneath. In the middle of the complexity, a window suddenly appears with a regular period similar three or 7 as a result of mode locking. The menstruation-3 bifurcation occurs at r=1+2sqrt(2)=3.828427..., and period doublings and then begin once again with cycles of vi, 12, ... and 7, fourteen, 28, ..., and then one time once again break off to anarchy. Notwithstanding, annotation that considerable construction can exist found inside this chaos (Mayoral and Robledo 2005ab).

It is relatively piece of cake to show that the logistic map is chaotic on an invariant Cantor ready for r>2+sqrt(5) approx 4.236 (Devaney 1989, pp. 31-50; Gulik 1992, pp. 112-126; Holmgren 1996, pp. 69-85), only in fact, information technology is as well chaotic for all r>4 (Robinson 1995, pp. 33-37; Kraft 1999).

The logistic map has correlation exponent 0.500+/-0.005 (Grassberger and Procaccia 1983), chapters dimension 0.538 (Grassberger 1981), and information dimension 0.5170976 (Grassberger and Procaccia 1983).

The logistic map can exist used to generate random numbers (Umeno 1998; Andrecut 1998; Gonzáles and Pino 1999, 2000; Gonzáles et al. 2001ab; Wong et al. 2001, Trott 2004, p. 105).

See also

2x modern one Map, Bifurcation, Feigenbaum Constant, Logistic Map--r=2, Logistic Map--r=2, Logistic Map--r=four, Period Three Theorem, Quadratic Map, Argent Constant, Tent Map

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References

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Logistic Map

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Weisstein, Eric W. "Logistic Map." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogisticMap.html

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