Scrambling at the genesis of chaos

Abstract

The presence of chaos in classical Hamiltonian systems is witnessed by its maximal Lyapunov exponent, that quantifies the instability of motion through the exponential growth of indicators such as the trace of the stability matrix or the out-of-time-ordered correlator. On the other hand, integrable dynamics near unstable fixed points, which are in turn characterized by a stability exponent, can also induce such exponential growth. Following the paradigm of integrability-breaking as driven by nonlinear resonances that hallmarks the genesis of chaos, the integrability-chaos transition is universally described by a periodic perturbation applied to a generic pendulum. Remarkably, this means that within the corresponding separatrix dynamics, which is an unavoidable a consequence of the resonance scenario, both instability exponents must play a role as both dynamical regimes coexist. We report here the universality of the transition from instability to Lyapunov exponents, thus completing the resonance scenario at the level of indicators based on exponential growth. To achieve this goal we obtain an analytical expression for the time evolution near separatrices, which enables us to derive an analytical expression for the exponent that characterises chaos and its transition from local instability to global chaos. We support our claim for the universality of this mechanism by studying two paradigmatic examples of the integrability-to-chaos transition, namely the kicked rotor and the driven pendulum.

Figures (8)

• Figure 1: OTOC growth in integrable, near-integrable and mixed systems. Left column: Phase-space representations of a) the simple pendulum given by \ref{['eq_hamiltonian']} with $\kappa=0$, c) the kicked Bose-Hubbard dimer given by \ref{['eq_kickedDimer']} 100 particles, $U=J$, and the period and strength of the hopping kick $\tau = 0.2 J$ and e) the kicked rotor given by\ref{['eq_kickedRotor']} with kicking parameter $K=0.6$. The center of the initial coherent state is indicated by a cross and corresponds to an unstable fixed point. Right column: quantum (red) and classical (blue) OTOC for b) the simple pendulum with $\hbar=2^{-11}$, d) the kicked dimer with $\hbar=1/3000$ and f) the kicked rotor with $\hbar=2^{-14}$. The operators used for the OTOC are shown in the label of the vertical axis. The straight lines represent exponential growth, with the rate given by $2\lambda_{\rm s}$ (cyan), $\lambda_{\rm s}$ (dashed cyan), and $(\lambda_{\rm s}+\lambda_{\text{L}})/2$ (long-dashed cyan).
• Figure 2: Closest position to the origin reachable before the homoclinic region formalism deviates from the numerical solution, plotted as a function of the inverse perturbation parameter $1/\kappa = \omega \lambda_{\rm s}/K$, obtained numerically (blue) and with the Melnikov method detailed in Appendix \ref{['app_sec_melnikov']} (darker shade of blue). The points were obtained using, $\kappa \omega = \lambda_{\rm s}$, $\varphi=0.1$. In solid purple, we show the closest position to the origin set by the linearisation condition of equation \ref{['eq:qdot']}, and in dashed purple an arbitrary scaling of $q_{\rm c} = \kappa^\alpha$, with $\alpha=1/2$, that satisfies both boundaries.
• Figure 3: First row: stroboscopic section of the driven pendulum. Second row: trace of the stability matrix for different perturbation parameters. Third row: Exponential growth rate extracted from the second row. Different rates are show as horizontal lines: $\lambda_{\rm s}$ in blue, $\lambda_{\rm s}/2$ in dashed blue, $(\lambda_{\text{L}}+\lambda_{\rm s})/2$ in orange, and $(\lambda_{\text{L}}+\lambda_{\rm s})/4$ in dashed orange. The last two rows use the same colour scheme: grey for the numerical solution of the simple pendulum, purple for the driven pendulum, and orange for the driven pendulum using the mapping derived in \ref{['sec_analyticalDescription']}. The first column is for $\kappa=0.0354$, the second one for $\kappa=0.0894$
• Figure 4: Exponential growth rates. The stability exponent $\lambda_{\rm s}$ is shown in orange, the effective exponent of the trace $\lambda_{\text{num}} = (\lambda_{\rm s}+\lambda{_\text{L}})/4$, obtained numerically is shown in red, $\lambda_{\text{num}}$ but with both exponents obtained using the Wolf algorithm in green, equation \ref{['eq:lambda2']}, labelled $\lambda_{\text{analytical}}$ is shown in purple using $\beta=20$, and \ref{['eq:lambda2alt']} in dashed purple, using $q_{\text{c}}=\kappa^{1/2}$ for the last two.
• Figure 5: First colum: Standard map/kicked rotor. Top: Phase space representation of the standard map for $K=0.6$, $\hbar_{\text{eff}} = 2^{-14}$. The initial coherent state is shown in red, centred at $(0,0)$. Bottom: quantum (red) and classical (blue) OTOC with the operator $\hat{A} = \hat{B} = \hat{p}$. Different exponential function are shown to highlight the different regimes. Second column: kicked dimer. Top: Phase space representation of the kicked dimer for $N=3000$, $U=0.025J$. The initial coherent state is shown in red. Bottom: quantum (red) and classical (blue) OTOC with the operator $\hat{A} = \hat{n}_1$, $\hat{B} = \hat{n}_2$. Different exponential function are shown to highlight the different regimes.