Abstract
We continue our exploration of systems without characteristic scales and specific methods into the field of dynamical systems, with the analysis of chaotic and turbulent behaviours. Due to the sheer magnitude of this field, our presentation will be deliberately selective, focusing only on critical aspects and behaviours and associated scaling laws. A brief introduction to the theory of dynamical systems takes us first of all to the important concept of bifurcation, a qualitative change in the asymptotic dynamics.
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- 1.
Let us emphasize at the start that throughout this chapter, “trajectories” are trajectories in phase space not in real space (unless the two spaces coincide).
- 2.
There is an important difference to understand intuitively between continuous and discrete dynamical systems. Trajectories of an autonomous (i.e. where V does not depend explicitly on time) continuous dynamical system can only cross or meet themselves at fixed points, which they only reach asymptotically. This topological constraint does not exist for discrete trajectories generated by a transformation. Given a discrete dynamical system in d dimensions, continuous dynamical systems generating dynamics whose trace obeys this discrete dynamical system take place in a dimension strictly larger than d, as explicitly shown in Fig. 9.2.
- 3.
The intuitive concept of an attractor \(\mathcal{A}\subset \mathcal{X}\) has appeared in various mathematical formulations, more of less strict depending on the context and the authors. They share the condition of invariance (\(f(\mathcal{A}) = \mathcal{A}\) in discrete time) and the fact that \(\mathcal{A}\) “attracts” (strictly it is the limit ensemble) all or some of the trajectories passing nearby. This means that the study of asymptotic behaviour of trajectories becomes the study of that of the dynamics restrained to \(\mathcal{A}\). See [59] for an in depth discussion of the concept of attractor and the different definitions proposed.
- 4.
Note that the analogy is with phase transitions in classical thermodynamics, described in a “mean field theory” in the sense of neglecting fluctuations and where only the average order parameter is written: the “critical exponents” of a bifurcation are always rational.
- 5.
Other notable normal forms exist, but are associated with non generic bifurcations (the associated bifurcation theorems involve equalities not inequalities):– transcritical bifurcation: \(\dot{x} = \mu x - {x}^{2}\); – pitchfork bifurcation: \(\dot{x} = \mu x - {x}^{3}\) for the supercritical case or \(\dot{x} = \mu x + {x}^{3} - {x}^{5}\) for the subcritical case (Fig. 9.5).
- 6.
Take a model depending on a real parameter a and a phase space \(\mathcal{X}\). The statement “for a < a 0, the solution …” is generic whereas the statements “for a = a 0, the solution …” or “for a ≤ a 0, the solution …” are not generic since a small variation in a changes the hypothesis to another hypothesis for which the statement is no longer true.
- 7.
This boils down to introducing a collective variable that destabilises at the instability threshold and whose behaviour therefore dominates the dynamics (general result the details of which we will give in Sect. 9.2.4).
- 8.
The oscillations and/or their coupling must be nonlinear; two ideal springs (harmonic oscillators) in series are equivalent to a single spring whose behaviour is perfectly predictably that of a harmonic oscillator.
- 9.
The derivation of this system of differential equations from the (spatiotemporal) hydrodynamic equations describing the evolution of the atmosphere can be found in [6].
- 10.
They are spontaneous in the sense that observed temporal variations are not simply the reflection of external temporal variations (for example, a periodically varying rate of injection of reagents).
- 11.
Let us emphasise that the reactor here is continuously fed (to maintain the concentrations of reagents constant) and agitated to ensure spatial homogeneity and avoid the formation of structures (incidentally very interesting and also much studied).
- 12.
By this we mean that the product of the reaction activates the enzyme and increases the reaction rate and therefore the product formation and so on, if there is no other mechanism capable of consuming the product.
- 13.
It is not relativistic effects that explain this unpredictability. Nevertheless, consequences of relativistic effects, as well as influences of bodies outside of the solar system, can be amplified due to the chaotic nature of the motion and the associated sensitivity to perturbations.
- 14.
These are reversible so they can be used to “go back” in time.
- 15.
This qualification is linked to the fact that the recorded signal Z(t) is generally scalar, whereas the phase space \(\mathcal{X}\) can be of dimension higher than 1. Therefore a procedure has to be used to reconstruct the trajectory \(z(t) \in \mathcal{X}\) from which the signal is derived (Z(t) = ϕ[z(t)] where ϕ is the measure function). The underlying idea is that each variable is affected by the set of other variables and so contains information about the global dynamics of the system. The most common procedure used is the method of delays, where the trajectory is reconstructed in n + 1 components: \([z(t),z(t - \tau ),\,\ldots \!\,,z(t - n\tau )]\). A discussion of this procedure, in particular of the choice of parameters τ and n is in [1, 21] and the original article [80].
- 16.
More precisely it is stated: For each trajectory starting from a point x 0 belonging to a subset of \({\mathcal{X}}_{0}\) of \(\mathcal{X}\) of full measure (that is to say such that \({m}_{\infty }(\mathcal{X}-{\mathcal{X}}_{0}) = 0\)) and for each observable F, we have: \({\lim }_{t\rightarrow \infty }{ 1 \over t} {\int\nolimits \nolimits }_{0}^{t}F({\phi }_{s}({x}_{0}))ds ={ \int\nolimits \nolimits }_{\mathcal{X}}F(x)d{m}_{\infty }(x)\).
- 17.
We emphasize that ergodicity is a property of the pair formed by the motion ϕ t and the invariant measure m ∞ . Nevertheless, we often use the short hand “ergodic measure” or “ergodic motion” when there is no ambiguity over the partner.
- 18.
A remarkable property of mixtures of different types of grains is that any vibration or rotation motion leads to segregation of the different types of grain. The homogeneity of the mixture cannot be improved by shaking, as we do for a suspension or emulsion. The idea is to include the grains to be mixed in a neutral paste, which is subjected to the baker’s transformation. The paste is then removed or a paste that does not interfere with the subsequent use of the granular mixture is used.
- 19.
We can always write points (x, y) in the square [0, 1] ×[0, 1] in the following form (dyadic expansion): \(x ={ \sum\nolimits }_{n=0}^{\infty }{2}^{-(n+1)}{\sigma }_{n}\) and \(y ={ \sum\nolimits }_{n=1}^{\infty }{2}^{-n}{\sigma }_{-n}\) where σ n = 0 or 1. It can be shown that B acting on (x, y) ends up shifting the indices of the set [σ]. The set [σ′] associated with B(x, y) is given by \({\sigma }_{n}^{{\prime}} = {\sigma }_{n+1}\) (shift operator). It then follows that the points (x, y) for which \({\sigma }_{n+N} = {\sigma }_{n}\) for all integer ratios of n and N (arbitrarily fixed), will have a periodic trajectory of period N under the action of B.
- 20.
Note that this projection in the unstable direction is the only one leading to a reduced dynamics that is deterministic and closed. The projection operation nevertheless transforms the underlying reversible dynamics into an irreversible reduced dynamics [19].
- 21.
This relation ε t = ε0eγt is approximate: it is not valid at short times, due to the influence of the transient regime and the local characteristics of the dynamics; nor is it valid at long times if the phase space (or the attractor if there is one) is bounded which “folds” the trajectories. It cannot therefore be considered as the definition of a Lyapounov exponent, which is a global and asymptotic quantity, but only as a simple and intuitive interpretation of this quantity.
- 22.
We should add a technical condition requiring that the law of motion f is continuously differentiable and Hölder-continuous.
- 23.
The exact definition of a strange attractor is a compact attractor containing a “homocline” trajectory (or orbit), that is to say a trajectory emerging from a point called a “homocline point” situated at the intersection of the stable manifold and the unstable manifold of a saddle type fixed point [36]. The dynamical complexity of such trajectories, in which it can be easily shown that all the points are homocline, had been already highlighted by Poincaré [23].
- 24.
The degree of irrationality of a real number r ∈ [0, 1] can be quantified by studying its rational approximations. For each integer q, we denote by p q, r ∕ q the best approximation of r by a rational number of denominator q. We can then define the subsets \({\mathcal{F}}_{\alpha }\) of [0, 1], containing the real numbers r such that \(\vert r - {p}_{q,r}/q\vert \leq{q}^{-\alpha }\) for an infinite number of integers q. It can be shown (Dirichlet’s theorem) that \({\mathcal{F}}_{2} = [0, 1]\) and that for all α > 2, \({\mathcal{F}}_{\alpha }\) is a fractal of dimension 2 ∕ α (Jarnik’s theorem). These sets are nested: \({\mathcal{F}}_{{\alpha }_{2}} \subset {\mathcal{F}}_{{\alpha }_{1}}\) if α1 < α2. The larger α, the more the elements of \({\mathcal{F}}_{\alpha }\) are “well approximated” by the rational numbers. By determining which sets \({\mathcal{F}}_{\alpha }\) the ratios ω1 ∕ ω n , …, \({\omega }_{n-1}/{\omega }_{n}\) belong to, we determine the order in which the associated invariant tori will disappear [24].
- 25.
The mathematical difficulty of the theorem is to define the term “in general” which requires considering a space of dynamical systems and endow it with a topology [20]. The structural stability of a quasiperiodic motion \(t \rightarrow\Phi ({\omega }_{1}t + {\varphi }_{1},\ldots,{\omega }_{n}t + {\varphi }_{n})\) with n periods depends on the perturbations considered. There is a structural instability from n = 3 if the perturbations are only constrained to be continually differentiable twice (class \({\mathcal{C}}^{2}\)), whereas the instability appears from n = 4, if only perturbations that are infinitely differentiable are allowed (class \({\mathcal{C}}^{\infty }\)). The quasiperiodic regime can therefore be replaced by a strange attractor.
- 26.
The arguments remain qualitatively valid for simple liquids but their technical implementation will be different, since the approximations allowed by the low density of gases could no longer be made.
- 27.
Note however that it behaves with a type of chaos a bit different from that presented in Sect. 9.2: the number of degrees of freedom is very large, whereas a characteristic of deterministic chaos is that it takes place in systems in low dimension. However this characteristic is not exclusive and molecular chaos seems to be an example of spatiotemporal extension of low dimensional chaos, involving the same ingredients: sensitivity to initial conditions, mixing and the existence of periodic trajectories of all periods [30].
- 28.
Another way of formulating the same point involves the concept of statistical ensemble: the temporal average is equal to the average over a large number of independent systems, constructed identically to the original system. This formulation, introduced by Gibbs, corresponds exactly to the actual concept of statistical sampling. In practice, what we call a “statistical ensemble” is an ensemble of microscopic configurations weighted by a probability distribution such that the statistical averages ⟨A⟩ correctly estimate the observed quantities A obs in the system under consideration. The situations presented above correspond to the microcanonical ensemble and the canonical ensemble respectively.
- 29.
This hypothesis is rarely explicitly justified for two reasons: it leads to theoretical results in agreement with experiments and it is not known how to prove this ergodicity in general.
- 30.
The term “out of equilibrium” is ambiguous and we should distinguish:
-
Systems relaxing towards their equilibrium state, sometimes slowly and in a complex way if metastable states exist.
-
Systems that have reached a steady state that is out of equilibrium, in the sense that there non zero flows (of material, energy etc) across them. We use the term “far from equilibrium” for these systems to distinguish them from the previous case.
-
- 31.
If w is the linear form associated with the eigenvalue 1 of the stability matrix, it is necessary that \(w[{D}_{\nu }{g}_{{\nu }_{c}}({X}^{{_\ast}})] > 0\) and \(w[{D}_{x}^{2}{g}_{{\nu }_{c}}({X}^{{_\ast}})] > 0\).
- 32.
A conjugacy is the replacement of \({g}_{\nu }\) by ϕ ∘ g ν ∘ ϕ − 1 and x by ϕ(x) where ϕ is an adequate diffeomorphism.
- 33.
It is even the key to the demonstration of bifurcation theory.
- 34.
The exponent \(-1/2\) is associated with the quadratic form of the evolution law. The family \({(-\mu+ x - A{x}^{1+\epsilon })}_{\mu }\) has a similar behaviour, with in this case \(\tau (\mu ) \sim{\mu }^{-\epsilon /(1+\epsilon )}\). It represents another universality class (where ε = 1 corresponds to the class of the classic saddle-node bifurcation).
- 35.
On-off intermittency is also encountered in a system of two coupled chaotic oscillators passing in an irregular manner from a synchronous motion, where the difference \(x = {X}_{1} - {X}_{2}\) between the states of the two oscillators cancels out, to an asynchronous motion where x≠0 [28, 68]. Here we observe self-similarity of the signal x(t) and the behaviour \(P(\tau ) \sim{\tau }^{-3/2}\) of the distribution of the duration τ of the synchronous phases (state “off” x = 0, analogue of the scarce phases) characterising this type of intermittency.
- 36.
As for the historical example of the Lotka–Volterra model, observation of fish in particular could be made for “practical” reasons: fishing by trawling provides representative sampling and accurate data are available over long periods of time from auction sale records.
- 37.
To take into account the case where individuals survive after reproduction, the model can be modified by introducing survival rates s 1 and s 2 of the two species:
$$ \left \{\begin{array}{l} x(t + 1) = x(t)({s}_{1} +\exp [{r}_{1} - {a}_{1}x(t) - {a}_{2}y(t)]) \\ y(t + 1) = y(t)({s}_{2} +\exp [{r}_{2} - {a}_{2}x(t) - {a}_{1}y(t)]). \end{array} \right. $$The results obtained with this second model prove to be qualitatively and even quantitatively (same exponent \(-3/2\)) identical to those presented for the first model.
- 38.
It consists exactly of a transverse Lyapounov exponent describing the stability within the global attractor of the solution y 0(t) corresponding to the resident species alone with respect to a transverse perturbation.
- 39.
Such a sequence leading first to chaos then to fully developed turbulence is also observed, in an exemplary way, in Rayleigh–Bénard’s experiment described in Fig. 9.1. However the phenomenon is different because in this case the density of the fluid varies as a function of its temperature. The dimensionless number controlling the fluid dynamics is no longer the Reynolds number but the Rayleigh number, involving the vertical gradient of temperature imposed on the system.
- 40.
Note that it consists of relative eddies, whose motion is described with respect to eddies of larger size in which they are embedded.
- 41.
An intermediate result, the Wiener–Khinchine theorem, relates the spectrum E(k) to the spatial correlation function of the velocity field according to the following formula: \(E(k) ={ \int\nolimits \nolimits }_{0}^{\infty }kr\;\sin (kr)\;\langle v(r + {r}_{0})v({r}_{0})\rangle \;dr/\pi \) where \(\langle v(r + {r}_{0})v({r}_{0})\rangle\) only depends on r through the statistical isotropy and homogeneity of turbulence.
- 42.
To be more rigorous, multifractal analysis should be done on the cumulative probability distribution: Prob\([\delta v(r,l,u) > {l}^{\alpha }]\). We can also develop a multifractal analysis of the dissipation \({\epsilon }_{l}(r)\), as it happens related to that of the velocity field because \(\delta v(l) \sim{(l{\epsilon }_{l})}^{1/3}\), so that \({\epsilon }_{l}(r) \sim{l}^{3\alpha (r)-1}\) [27].
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Lesne, A., Laguës, M. (2012). Dynamical Systems, Chaos and Turbulence. In: Scale Invariance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15123-1_9
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