Abstract
The theory of chaos in one-dimensional maps is described. We start with the basic bifurcations, following which there is an extended discussion of period-doubling sequences and the Feigenbaum conjectures. There follows an extended discussion of kneading theory, culminating with a statement of Sarkovskii’s ordering of the periodic orbits of unimodal maps.
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Notes
- 1.
Because of this, (2.3) is not wholly satisfactory; a better kind of function has \(f(x) \rightarrow 0\) as \(x \rightarrow \infty .\) One example which has been used is \(x_{n + 1} = \lambda x_{n} (1 + x_{n})^{- \beta }\), with \(\beta >1\).
- 2.
Since \(x_n<x_{n-1}\) for all n.
- 3.
\(x=y\ \mathrm{mod}\ z\) if \((x-y)/z\) is an integer.
- 4.
The sequences \(\{x_{n}\}\) may also be determined from the tent map \(\theta _{n+1} = 2 \theta _{n}, \ 0< \theta _{n}< \pi , \ \theta _{n+1} = 4 \pi - 2 \theta _{n}, \ \pi< \theta _{n} < 2 \pi \) (see Chap. 1).
- 5.
Providing we exclude sequences ending \(\ldots 1111\ldots \).
- 6.
Trajectories which hit \(\phi = \frac{1}{2}\) are those which terminate \(\ldots 10000 \ldots \), since if \(\phi _{k} = \frac{1}{2}\), then \(\phi _{k} = 1, \phi _{k+i} = 0\) for \(i \ge 1\).
- 7.
And then, one needs a symbol C to denote when \(\phi _{k} = c\), where c is the turning point of f.
- 8.
In his book, An Introduction to Chaotic Dynamical Systems.
- 9.
There are actually two others, the transcritical bifurcation and the pitchfork bifurcation (cf. question 1.3): in both of these \(f_\mu =0\) at the fixed point \(x^*\), and in the second also f is odd about \(x^*\); see question 2.6.
- 10.
- 11.
A manifold is simply a space that is locally Euclidean, i.e. it looks like \(\mathbf{R }^n\). For example, the surface of a sphere is a two-dimensional manifold. In a function space, it is most easily conceived as a subspace of functions such as \(\sum _1^na_if_i(x)\), or more generally a family of functions f(x, a), where for an n-dimensional manifold, \(a\in \mathbf{R }^n\). A one-dimensional unstable manifold for the fixed point g is then an invariant manifold on which the points are mapped away from g by the map T.
- 12.
Transverse here has an obvious geometrical meaning, and is as illustrated in Fig. 2.10. Provided we have concepts of angles and tangents, we want the angle of the tangents at the point of intersection of \(W^u\) and \(\Lambda _1\) to be non-zero. Angles are interpreted in function spaces by inner products, and tangents by linear operators known as Frêchet derivatives.
- 13.
The co-dimension of a subspace \(\mathbf{R}\!^k\in \mathbf{R}\!^n\) is just \(n-k\), the dimension of the complement. More generally we talk of the co-dimension of a manifold. The use of the term becomes advantageous when, as here, we are considering subspaces of infinite-dimensional spaces.
- 14.
In his book, Sparrow (1982) uses the symbols L, R instead of 0, 1.
- 15.
The discrepancy \(n\ge 0\) is the first integer in the sequences (2.31) for \(S(x)=\mathbf{{s}}\) and \(S(y)=\mathbf{{t}}\) for which \(s_n\ne t_n\).
- 16.
- 17.
We have strayed deep into the land of pure mathematics. Only a pure mathematician would imagine proving a constructive result by showing that a set is open.
- 18.
A superstable periodic orbit \(\{x_i\}\) is one which passes through c, since then the quantity \(\prod _if'(x_i)=0\).
- 19.
See Devaney (1986), remark 2 on p. 146, for example.
- 20.
A full statement of the implicit function theorem is coming up shortly, at the beginning of Chap. 3. In the present context, it says that if the function \(g(x,\varepsilon )\) satisfies \(g(x^*, 0)=0\) and has non-zero derivative (with respect to x) at the same point, and is smooth (continuously differentiable), then for small y, the equation \(g(x,\varepsilon )=y\) defines a smooth function \(x=X(y,\varepsilon )\); see, e.g., Devaney’s book.
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Fowler, A., McGuinness, M. (2019). One-Dimensional Maps. In: Chaos. Springer, Cham. https://doi.org/10.1007/978-3-030-32538-1_2
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DOI: https://doi.org/10.1007/978-3-030-32538-1_2
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