Abstract
Dynamical systems theory offers a different, geometrical view for describing the evolution of a system. It leads to the puzzling notions of deterministic chaos and universal routes to chaos.
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Notes
- 1.
The real system \(\mathcal{S}\) and the dynamical system modeling the evolution of a few quantities characterizing the state of \(\mathcal{S}\) should not be confused.
- 2.
Rigorously, V is defined on \(\mathcal{X}\) and its value V(x) belongs not to \(\mathcal{X}\) but to the tangent space to \(\mathcal{X}\) in x; the distinction is important for instance when \(\mathcal{X}\) is a circle and x an angular variable. In mathematical terms, such a function V is called a vector field on \(\mathcal{X}\).
- 3.
It is enough that V is \(C^1\) in \(\mathcal{X}\), that is, differentiable with a continuous differential \(x\mapsto DV(x)\), to ensure the existence and unicity of the solutions. The rationale behind such a regularity assumption is that we are interested in physical models intending to account for realistic dynamics. Singular evolution equations may have several solutions for a given initial conditions, or even worst, no solutions. We here exclude such unphysical behaviors.
- 4.
In order to better understand this theorem, the reader can compute the determinant of the Jacobian matrix of the transformation \((q,p)\mapsto (q+\dot{q}\varDelta t,p+\dot{p}\varDelta t)\) which approximately describes (at the lowest order in \(\varDelta t\)) the evolution over a time step \(\varDelta t\). The rigorous proof for the exact evolution law (in continuous time) yields the same conclusion.
- 5.
\(a\equiv 0\) is a solution. As any trajectory, it behaves as an impenetrable barrier trapping the other trajectories either in the region \(a>0\) or in the region \(a<0\) according to the initial condition. This reasoning extends to the other variables.
- 6.
The statement is a bit more complicated in infinite dimension, since then the spectrum of \(DV(x^*)\) does not reduce to eigenvalues and the absence of eigenvalues with a vanishing real part does not guarantee the invertibility.
- 7.
If \(DV(x^*)\) has an eigenvalue of multiplicity \(m>1\) with only one associated eigenvector, it is not diagonalizable and can only be reduced to a Jordan form. This restriction does not markedly change stability properties since the ensuing modification of (6.17) is the addition of terms \(t^\mathrm {j}e^\mathrm {\eta t}\) (\(j=1,\ldots ,m-1\)) to the exponential \(e^\mathrm {\eta t}\).
- 8.
If \(E=U_{\max }\), the particle comes close to the unstable state \(x_{\max }\), such that \(U(x_{\max })=U_{\max }\), however with a velocity tending to 0; the particle will need an infinite time before it actually reaches \(x_{\max }\), and it will never go beyond. In any case, this situation has no practical interest because it is destroyed by the slightest perturbation, which turns it in one of the situations \(E<U_{\max }\) or \(E>U_{\max }\); in contrast, those latter situations are robust with respect to a small enough change in E.
- 9.
Strictly, one should distinguish the height \(U^+_{\max }\) of the barrier on the right of the well \(x_{\min }\) and the height \(U^+_{\max }\) of the barrier on the left of the well \(x_{\min }\), and consider the sign of the initial velocity.
- 10.
In the case of small oscillations around \(x^{*}\), the force \(-U^{\prime }(x)\) may be reduced to its linear approximation \(U^{\prime \prime }(x^{*})(x-x^{*})\), which amounts to replace U(x) by its harmonic approximation \(U(x)\approx U(x^{*})+ U^{\prime \prime }(x^{*})(x-x^{*})^2/2\) . Solutions of the linearized evolution are of the form \(A\cos (\omega t+\varphi )\) where the constants A and \(\varphi \) are determined by the particle position and velocity at the initial time.
- 11.
However, this statement is wrong in the special case where \(U^{\prime }\) vanishes on an interval. The state in which the system actually stops, among all the possible equilibrium states, would then depend on the system history and the friction it has experienced.
- 12.
Note that this initial state may be far different from the fixed point: often the basin of attraction extends up to infinity in some directions, and it may even sometimes coincide with the whole phase space.
- 13.
However the flow is here abstract, in the phase space, whereas it is composed of real water, flowing in the real space, in hydrodynamics.
- 14.
Conservative systems are a different case, which will be treated in Sect. 6.6.4, with the presentation of KAM theorem; this theorem describes a behavior that can be interpreted as the analogue of a bifurcation for Hamiltonian systems.
- 15.
As in the case of continuous time, if \(Dg(x^*)\) is not diagonalizable and possesses a Jordan block \(k\times k\) associated with the eigenvalue \(\varLambda \), terms \(n\varLambda ^\mathrm {n},\dots ,n^\mathrm {k-1} \varLambda ^\mathrm {n}\) appear. They do not affect the stability properties of the fixed point.
- 16.
The basic argument is: \(\mathrm {Det}(A_1A_2-\lambda {\mathbf {1}})=\mathrm {Det}(A_2A_1-\lambda {\mathbf {1}})\) hence \(A_1A_2\) and \(A_2A_1\) have the same eigenvalues, even if \(A_1A_2\ne A_2A_1\).
- 17.
On the contrary, the “naive” discretization \(g(x)=f_{\Delta {\mathrm {t}}}(x)\) where the time step \(\varDelta t\) is identical for all trajectories and all steps yields a discrete dynamical system in \(\mathcal{X}\).
- 18.
The continuous dynamical system has an additional eigenvalue \(\lambda _0=0\) corresponding to the invariance upon a translation by an integral number of turns along the cycle. The reader can easily check this statement on Example 7.
- 19.
Strictly, like V[x(t)], \(\varDelta x(t,x_0,\varDelta x_0)\) belongs to the tangent space of \(\mathcal{X}\) in \(x(t)=f_\mathrm {t}(x_0)\).
- 20.
Strictly, one should rather speak of the “invariant measure” m of the dynamical system and writes the infinitesimal volume element \(\mathrm {d}m(x)\). It can be written \(\mathrm {d}m(x)=\rho (x)\mathrm {d}x\) as soon as m possesses a smooth density.
- 21.
The exponents \((\gamma _\mathrm {j})_\mathrm {j\ge 1}\) in general differ from the eigenvalues \((\lambda _\mathrm {j})_\mathrm {j\ge 1}\), unless the flow as a unique stable fixed point with a basin of attraction equal to the whole phase space \(\mathcal{X}\). Indeed in this case, trajectories rapidly reach a neighborhood of \(x^*\) where DV[x(t)] and \(DV(x^*)\) can be identified.
- 22.
To simplify, we consider the simplest case where the measure is defined by a density: \(\mathrm {d}m(x)=\rho (x)\mathrm {d}x\). It is not always the case, and strictly all should be written using the volume element \(\mathrm {d}m(x)\).
- 23.
A flow, with invariant measure m, is ergodic if any invariant subset \(\mathcal{B}\) of \(\mathcal{X}\) (\(f(\mathcal{B}\subset \mathcal{B}\)) either has null measure (\(m(\mathcal{B})=0\)), or its complementary has null measure (\(m(\mathcal{X}-\mathcal{B})=0\)). This property of the pair (f, m) is best understood using the equivalent definition of ergodicity: a typical trajectory visits an infinite number of times any region \(\mathcal{B}\) of \(\mathcal{X}\) provided the measure of \(\mathcal{B}\) does not vanish (even if it is arbitrary small).
- 24.
A mode is a component \(c_{\lambda }\mathrm {e}^\mathrm {\lambda t}\) of an evolution f(t). The coefficient \(c_{\lambda }\) is the amplitude of the mode; it is related to the Fourier transform of f (if \(t\in ]-\infty ,+\infty [\) and f integrable) of the Laplace transform of f (if \(t>0\)). The mode is stable if \(\mathfrak {R}(\lambda )<0\), unstable if \(\mathfrak {R}(\lambda )>0\) (for \(t\rightarrow +\infty \)).
- 25.
If \(\epsilon >0\), the tori are deformed: they are termed tori as long as the deformed set remains smoothly related (diffeomorphic) to the torus \(\{J_1,\ldots ,J_\mathrm {d}\}\times {\mathbf {T}}^\mathrm {d}\) observed for \(\epsilon =0\).
- 26.
Dubbed from the greek mathematician Diophantus of Alexandria (325–410).
- 27.
Acknowledged for a mundane theorem that was besides established earlier, Pythagoras (560–480) was also a mystical philosopher; we owe him the first results of number theory, in particular the irrational nature of \(\sqrt{2}\).
- 28.
Hint: introduce an additional variable s, with \(s(t)\equiv t\).
- 29.
Hint: define a k-dimensional state variable \(X_\mathrm {n}=(x_\mathrm {n-k+1},\ldots ,x_\mathrm {n-1}, x_\mathrm {n})\).
- 30.
Hint: a qualitative change takes place for \(\gamma ^2=4Km\).
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Stauffer, D., Stanley, H.E., Lesne, A. (2017). Dynamical Systems and Chaos. In: From Newton to Mandelbrot. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53685-8_6
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