Chaos in Dissipative Systems

Abstract

A crucial distinction exists between the dissipative systems and conservative Hamiltonian systems. By Liouville’s Theorem, the solution flow for a conservative Hamiltonian system preserves volumes in phase space. Dissipative systems, by contrast, usually give rise to solution flows which contract volumes in phase space. This volume contraction gives rise to a bounded set called an attractor in the phase space, which ultimately contains the solution flow, once the transients associated with initial conditions have died away. Indeed, dissipative systems of dimensions equal to and greater than three can have bounded trajectories, which may be attracted not by a fixed point nor by a periodic/quasi-periodic orbit, but by an object of complicated infinitely many-layered structure called a strange attractor. The trajectories on this attractor diverge continually from each other locally, but remain bounded globally. Besides, the evolution on the attractor is essentially aperiodic. Strange attractors are sometimes modeled by fractals which are geometric objects that are very suited for characterizing roughness and illustrating the difference between the mathematical properties of continuity and differentiability. Adoption of fractal geometry therefore releases one from bondage to smooth surfaces and smooth curves and enables one to come to terms with nature, for as Mandelbrot (The Fractal Geometry of Nature, Freeman, 1983) put it—“clouds are not spheres, mountains are not (cones), and bark is not smooth, nor does lightning travel in a straight line”. (Indeed, the fractal structure of a natural object is closely intertwined with its functionality (like the light-catching capacity of trees or the electrical connectivity of neurons).) Further, fractals are self-similar objects which have the same shape at all scales.

Appendices

Appendix A: The Hausdorff-Besicovitch Dimension

Let us consider a measure of the size of a set of points, S, in space, corresponding to a resolution of the measurement δ. We take a function h(δ)≡γ(d)δ ^d, which may represent a line, square, disk, ball or cube, and cover the set S to form the measure:

$$ M_d ( S )=\sum h(\delta)= \sum\gamma( d)\delta^d. $$

(6.135)

Here, the geometrical factor γ(d)=1 for lines, squares and cubes; γ=π/4 for disks and γ=π/6 for spheres. In the limit δ→0, the measure M _d (S) turns out to be zero or infinite depending on the choice of d—the dimension of M _d (S). (Note that for the Koch curve, the 1-dimensional measure, namely, the length, is diverging, while the 2-dimensional measure, namely, the area is 0.) The Hausdorff-Besicovitch dimension D(S) of the set S is the critical dimension for which the measure M _d (S) jumps from zero to infinity:

(6.136)

or

$$ D ( S)=\operatorname{inf} \bigl\{ d|M_d( S)=0\bigr\} =\operatorname{sup}\bigl\{ d|M_d( S)=\infty\bigr\} $$

which implies N(δ)∼δ ^−D(S), or that the measure is finite, when d=D(S). Note that this definition makes the Hausdorff-Besicovitch dimension a local property in the sense that it measures properties of sets of points in the limit of a vanishing parameter or size δ of the test function used to cover the set.

Some fundamental properties of the Hausdorff-Besicovitch dimension may be noted:

• (i) if S⊂R ⁿ, then D(S)≤n;

• (ii) if S⊂U, then D(S)≤D(U);

• (iii) if S⊂R ⁿ, and D(S)<1, then S is totally disconnected.

Equation (6.136) also implies that an object may be called a fractal, if it is not possible to obtain a finite measure for its volume, surface or length when measuring with d,d−1,… dimensional hypercubes of size ε and changing ε over several orders of magnitude. Thus, the length estimate L(δ) of a fractal curve varies with the resolution δ as L(δ)∼δ ^1−D, where D is the dimension of the curve; D is clearly equal to unity for classical curves (in the limit of small δ) and is larger than unity for fractal curves.

Example 6.9

(The Cantor set)

Let A be the cover of the Cantor set consisting of 2ⁿ intervals of length 1/3′′ each, for n=1,2,…. Then, we have from (6.136),

$$ 2^n \cdot\biggl(\frac{1}{3^n}\biggr)^D=1 $$

from which, the Hausdorff-Besicovitch dimension D is

$$ D=\frac{\ln 2}{\ln 3}=0.69 $$

in agreement with the capacity dimension derived in Example 6.2.

Note that, for a given set, the Hausdorff-Besicovitch dimension need not be the same as the capacity dimension because, the latter does not distinguish between a set and its closure (Falconer 2003) and hence leads to counter-intuitive results. For example, the set of rational numbers is a countable union of points (of measure 0) so that its Hausdorff-Besicovitch dimension is 0 while its capacity dimension is 1 because its closure is R.³⁴ In general, the Hausdorff-Besicovitch dimension is smaller than (or equal to) the capacity dimension (Farmer et al. 1983).

Appendix B: The Derivation of Lorenz’s Equation

Consider a fluid layer of depth H, with the lower surface maintained at a higher temperature T ₀+ΔT than the temperature T ₀ of the upper surface. The equation governing the evolution of vorticity in this layer, with the inclusion of buoyancy effects is

$$ \frac{\partial}{\partial t}\nabla^2\psi+\biggl(\frac{\partial\psi}{\partial x} \frac{\partial}{\partial z}-\frac{\partial\psi}{\partial z}\frac {\partial}{\partial x}\biggr)\nabla^2 \psi-\nu\nabla^4\psi-g\alpha\frac {\partial\theta}{\partial x}=0 $$

(6.137)

where, the flow has been assumed to depend only on x (the horizontal direction), z (the vertical direction) and t. If v=(u,v) is the fluid velocity, the stream function ψ is defined by,

$$ u=-\frac{\partial\psi}{\partial z},\qquad v=\frac{\partial\psi}{\partial x}. $$

(6.138)

θ is the departure of the temperature profile from the conduction situation in the fluid,

$$ \theta=T-T_0-\Delta T\biggl(1-\frac{z}{H}\biggr). $$

(6.139)

g is the acceleration due to gravity, α is the coefficient of thermal expansion and ν is the kinematic viscosity of the fluid.

The equation governing the evolution of temperature in the fluid layer is

$$ \frac{\partial\theta}{\partial t}+\biggl(\frac{\partial\psi}{\partial x}\frac{\partial}{\partial z}- \frac{\partial\psi}{\partial z}\frac {\partial}{\partial x}\biggr)\theta-\frac{\Delta T}{H} \frac{\partial\psi }{\partial x}-\kappa\nabla^2\theta=0 $$

(6.140)

where κ is the thermal diffusivity of the fluid.

As the temperature difference across the fluid layer ΔT is increased, the heat transport process via the conduction mechanism becomes unstable, and gives way to the transport process via the convection mechanism involving a steady cellular flow. (The latter arises due to the fact that the fluid near the warmer lower surface expands and rises because of the buoyancy, while the cooler and heavier fluid near the upper surface descends under gravity.) If ΔT is increased further, the convection process shows a time-dependent behavior. In this regime, taking clues from the previous numerical work of Saltzman (1962), Lorenz (1963) considered the dynamics to be primarily described by three Fourier modes,

$$ \begin{aligned} \psi&=X( t)\frac{2^{1/2}(1+a^2)}{a^2}\sin \frac{\pi a x}{H}\sin \frac{\pi z}{H} \\ \theta&=Y( t )\frac{2^{1/2}R_c}{\pi R}\cos\frac{\pi a x}{H}\sin \frac{\pi z}{H}-Z( t)\frac{R_c}{\pi R}\sin\frac{2 \pi z}{H} \end{aligned} $$

(6.141)

where R is the Rayleigh number,

$$ R\equiv\frac{g \alpha H^3\Delta T}{\nu\kappa} $$

and R _c is the value of R, for which conduction gives way to convection. Upon substituting (6.141), equations (6.137) and (6.140) give Lorenz’s equations,

$$ \left . \begin{aligned} \dot{X}&=-\sigma( X-Y) \\ \dot{Y}&=rX-Y-XZ \\ \dot{Z}&=-bZ+ XY \end{aligned} \right\} $$

(6.142)

where the dot denotes derivative with respect to a non-dimensionalized time $\tau\equiv\frac{\pi^{2}( 1+a^{2})\kappa t}{H^{2}}$, and

$$ r\equiv\frac{R}{R_c},\qquad \sigma=\frac{\nu}{\kappa}\quad\text{and}\quad b = \frac{4}{1+a^2}. $$

Appendix C: The Derivation of Universality for One-Dimensional Maps

We will give here an intuitive derivation of universality for one-dimensional maps which have a single, locally quadratic maximum (the following formulation is adapted from Holton and May (1993)).

Consider the composite map

$$ x_{n+k}=f^{[k]}_\lambda( x_n). $$

(6.143)

Let $\overline{x}^{(k)}$ be the fixed point of this map; $\overline {x}^{(k)}$ is therefore a period-k cycle. Let,

$$ \mu^{(k)}(\lambda)\equiv\frac{\partial f^{[k]}_\lambda }{\partial x}\bigg| _{x=\overline{x}^{(k)}} $$

(6.144)

So, a stable period-k cycle appears at μ ^(k)=1, which becomes unstable, giving birth to a period-doubled period-2k cycle, at μ ^(k)=−1,³⁵ with μ ^(2k)=1. The latter then becomes unstable when μ ^(2k)=−1, and μ ^(k) takes some negative value $\mu^{(k)}_{c}$, say.

Let $\lambda^{(k)}_{0}$ be the value of the parameter λ, at which the period-k cycle appears, i.e.,

$$ \mu^{(k)}\bigl(\lambda^{(k)}_0\bigr)=1. $$

(6.145)

Writing,

$$ \lambda=\lambda^{(k)}_0+\varepsilon $$

(6.146)

a Taylor expansion then gives

$$ \mu^{(k)}(\lambda)=1+\varepsilon M^{(k)}_0 +O\bigl(\varepsilon ^2\bigr),\quad \lambda\approx\lambda^{(k)}_0 $$

(6.147)

where

$$ M^{(k)}_0 \equiv\frac{d \mu^{(k)}}{d \lambda}\bigg|_{ \lambda=\lambda ^{(k)}_0} . $$

Similarly, let $\lambda^{(2k)}_{0}$ be the value of λ at which the period-k cycle disappears, (and period-2k cycle appears), i.e.,

$$ \mu^{(k)}\bigl(\lambda^{2k}_0\bigr)=-1. $$

(6.148)

Writing,

$$ \lambda=\lambda^{(2k)}_0+\varepsilon $$

(6.149)

a Taylor expansion then gives

$$ \mu^{(k)}(\lambda)=-1+\varepsilon M^{(2k)}_0+O \bigl( \varepsilon^2\bigr),\quad \lambda\approx\lambda^{(2k)}_0 $$

(6.150)

where,

$$ M^{(2k)}_0\equiv\frac{d \mu^{(k)}}{d \lambda}\bigg|_{\lambda= \lambda ^{(2k)}_0}. $$

If Δλ(k) denotes the change in λ, as μ ^(k) varies from 1 to −1, then we have from (6.147),

$$ \Delta\lambda( k)=-\frac{2}{M^{(k)}_0}. $$

(6.151)

Similarly, if Δλ(2k) denotes the change in λ, as μ ^(2k) varies from 1 to −1 or μ ^(k) varies from −1 to $\mu^{(k)}_{c}$, then we have from (6.150),

$$ \Delta\lambda(2k)=\frac{\mu^{(k)}_c+1}{M_0^{(2k)}}. $$

(6.152)

Using (6.151) and (6.152), the Feigenbaum number δ (see (6.106) and (6.107)) is given by

$$ \delta=\lim_{k \rightarrow\infty}\frac{\Delta\lambda( k) }{\Delta\lambda(2k)}= \lim_{k \rightarrow\infty}\frac {-2}{\mu^{(k)}_c +1}. $$

(6.153)

In order to determine $\mu^{(k)}_{c}$, we need an asymptotic relation between μ ^(2k) and μ ^(k). For this purpose, we approximate the map $f^{[2k]}_{\lambda}$ by using a cubic:

$$ f^{[2k]}_\lambda\bigl(\overline{x}^{(k)}+\xi\bigr) \approx\overline {x}^{(k)}+ A\xi+\frac{1}{2} B\xi^2+ \frac{1}{6} C\xi^3+O\bigl(\xi^4\bigr). $$

(6.154)

The cubic approximation is necessary to enable $f^{[2k]}_{\lambda}$ to exhibit a bifurcation into a pair of points (corresponding to a period-2k cycle), as the period-k cycle becomes unstable.

Noting that

$$\begin{aligned} f^{[2k]}_\lambda\bigl(\overline{x}^{(k)}+ \xi \bigr) =&f^{[k]}_\lambda\bigl( f^{[k]}_\lambda \bigl( \overline{x}^{(k)}+\xi\bigr)\bigr) \\ {}=&f^{[k]}_\lambda\biggl( \overline{x}^{(k)}+ \mu^{(k)}\xi+\frac{1}{2} \sigma^{(k)}\xi^2+ \cdots\biggr) \end{aligned}$$

(6.155)

we have, on comparison with (6.154),

$$ A=\bigl(\mu^{(k)}\bigr)^2,\qquad B= \mu^{(k)}\bigl(1+ \mu^{(k)}\bigr)\sigma^{(k)} $$

(6.156)

where,

$$ \sigma^{(k)}\equiv\frac{\partial^2 f^{[k]}_\lambda}{\partial x^2}\bigg|_{x= \overline{x}^{(k)}}. $$

If we write for the fixed points $\overline{x}^{(2k)}$ of the map $f^{[2k]}_{\lambda}$,

$$ \overline{x}^{(2k)}=\overline{x}^{(k)}+ \xi^\ast $$

(6.157)

then we have

$$\begin{aligned} f^{[k]}_\lambda\bigl(\overline{x}^{(2k)} \bigr) =&f^{[2k]}\bigl(\overline {x}^{(k)}+\xi^\ast\bigr) \\ {}=& \overline{x}^{(k)}+A\xi^\ast+\frac{1}{2}B \xi^{\ast ^2}+\frac{1}{6}C\xi^{\ast^3}+\cdots= \overline{x}^{(k)}+\xi^\ast \end{aligned}$$

or

$$ \xi^\ast\approx A\xi^\ast+\frac{1}{2}B \xi^{\ast^2}+\frac{1}{6}C\xi ^{\ast^3}. $$

(6.158)

On discarding the degenerate period-k solution (ξ ^∗=0), (6.158) yields

$$ 0\approx( A-1)+\frac{1}{2}B\xi^{\ast}+ \frac{1}{6}C\xi^{\ast^2}. $$

(6.159)

Next, we have from (6.154),

$$ \mu^{(2k)}=\biggl[\frac{df^{[2k]}_\lambda}{d\xi}\biggr]_{\xi=\xi^\ast}\approx A+B\xi^\ast+\frac{1}{2}C\xi^{\ast^2}. $$

(6.160)

Using (6.159), (6.160) becomes

$$ \mu^{(2k)}\approx(3-2A)-\frac{1}{2}B \xi^\ast. $$

(6.161)

Dropping the second term on the right hand side in (6.161) in comparison with the first, we have corresponding to μ ^(2k)=−1 or $\mu^{(k)}=\mu^{(k)}_{c}$,

$$ -1\approx3-2A $$

or

$$ A\approx2. $$

(6.162)

Using (6.162), we have from (6.156),

$$ \mu^{(k)}_c \approx-\sqrt{2}. $$

(6.163)

Using (6.163), we have from (6.153),

$$ \delta=\frac{-2}{-\sqrt{2}+1}=4.828\ldots $$

(6.164)

which differs from the exact numerical result (equations (6.106) and (6.107)) by about 3 %!