Geometric investigation of chaos unfolding in Hamiltonian systems
Università di Padova（帕多瓦大学）；Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR)（意大利国家研究委员会光学研究所）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova（意大利国家核物理研究所帕多瓦分部）；Consorzio RFX (CNR, ENEA, INFN, Università di Padova, Acciaierie Venete SpA)（RFX联盟）
L. Salasnich, F. Sattin
发表信息：
Chaos, Solitons and fractals, vol. 208, art 118259 (2026)
In this work we revisit the geometric approach to chaos in Hamiltonian dynamics, by means of the Jacobi-Levi-Civita equation (JLCE). We inspect numerically two low-dimensional dynamical systems; show that, along chaotic orbits, the exponential divergence between nearby trajectories quantified by the JLCE does not unfold in a continuous manner, rather is closer to a multiplicative discrete process: in correspondence of each turning point, where the trajectory bounces away from the boundary of the energetically allowed region, the relative separation increases sharply and abruptly. We highlight through analytical and numerical arguments that the chaotic rather than regular nature of the trajectory is determined by the details of the scattering with the boundary, and interpret these results in terms of parametric resonance theory, and specifically the Mathieu equation.
Random matrix theory (RMT) provides a framework to study the spectral fluctuations in physical systems. RMT is capable of making predictions for the fluctuations only after the removal of the secular properties of the spectrum. Spectral unfolding procedure is used to separate the local level fluctuations from overall energy dependence of the level separation. The unfolding procedure is not unique. Several studies showed that statistics of long-term correlation in the spectrum are very sensitive to the choice of the unfolding function in polynomial unfolding. This can give misleading results re…
We study the origin of homoclinic chaos in the classical 3D model proposed by O. Rössler in 1976. Of our particular interest are the convoluted bifurcations of the Shilnikov saddle-foci and how their synergy determines the global unfolding of the model, along with transformations of its chaotic attractors. We apply two computational methods proposed, 1D return maps and a symbolic approach specifically tailored to this model, to scrutinize homoclinic bifurcations, as well as to detect the regions of structurally stable and chaotic dynamics in the parameter space of the Rössler model.
In this paper, we show that "Labyrinth walks", the conservative version of "Labyrinth chaos" and member of the Thomas-Rössler class of systems, does not admit an autonomous Hamiltonian as a constant function in time, and as a consequence, does not admit a symplectic structure. However, it is conservative, and thus admits a vector potential, being at the same time chaotic. This exceptional set of properties makes "Labyrinth walks" an elegant example of a chaotic, conservative, non-Hamiltonian system, with only unstable stationary points in its phase space, arranged in a 3-dimensional grid. As a…
Unfolded Krylov complexity: universal chaotic dynamics without false positives
Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Shanghai Jiao Tong University（上海交通大学）；Pohang University of Science and Technology（韩国科学技术院）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）
This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. More precisely, a Hamiltonian flow is identified with a geodesic flow on configuration space-time endowed with a suitable metric due to Eisenhart. Until now, this framework has never been given attention to describe chaotic dynamics. A gap that is filled in the present work. In a Riemannian-geometric context, the stability/instability of the dynamics depends on the curvature properties of the ambient manifold and is investigated by means of the Jacobi--Levi-Civita (JLC) equation for geodesic spread. It is…
It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a turbulent electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in partic…
We discuss Hamiltonian model of oscillator lattice with local coupling. Model describes spatial modes of nonlinear Schrödinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of Topaj - Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with dynamics exactly equivalent to the Topaj - Pikovsky model. We demonstrate the complexity of dynamics with results of numerical simulations. We also propose two dissipative models close to Topaj - Pikovsky system.
Probing Hamiltonian dynamics by means of the 0-1 test for chaos
最近提出的混沌测试[Gottwald GA and Melbourne I 2004 \textit{Proc. R. Soc. London A} { \bf 460} 603-611]被用来探测一维非谐振子晶格的哈密顿动力学。对于处于弱混沌状态的均匀（均匀质量）晶格和嵌入在晶格中的重杂质，在前一种情况下，由单个振子的位置和动量的时间记录产生的结果，以及在后一种情况下，对于与杂质对应的相同变量，不能确定系统的动力学状态。这种看似奇怪的行为有其根源，在不足的时间序列长度使用。然而，对于这两种情况，获得正确结果所需的必要时间记录长度使得测试不切实际。特别地，对于第二种情况，特别是在大的系统尺寸限制中（由于其与布朗运动的连接，这是物理上相关的限制），正确分类信号的测试所需的位置时间序列的估计长度超出了当今计算机能力的范围。因此，我们的研究结果表明，所提出的测试，上述情况下的哈密顿混沌，提供了没有明显的优势，比传统的相空间重构方法。
We perform a detailed study of the chaotic component in mixed-type Hamiltonian systems on the example of a family of billiards [introduced by Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)]. The phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times. The structure of the chaotic component is discerned from the cells visited by the chaotic orbit. The fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box-counting method. The cell-filling dynamics is compared to a model …
Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are not order one perturbations of a given Hamiltonian. This class of problems is very important for many applications, for instance in celestial mechanics. As an example, we study a class of one-dimensional Hamiltonians that depend explicitly on time and on stochastic external parameters. The variations of the external parameters are responsible for a distortion …
Recurrence time entropy and weak chaos in Hamiltonian flows
São Paulo State University (UNESP)（圣保罗州立大学）；Institute of Geosciences and Exact Sciences（地球科学与精确科学研究所）；University of São Paulo（圣保罗大学）；Institute of Physics（物理研究所）；State University of Ponta Grossa（蓬塔格罗萨州立大学）
The Madden-Julian Oscillation (MJO) has been studied using its daily RMM1, RMM2 and VPM1, VPM2 principal components' indices. A spectral analysis of the raw indices indicates presence of a noise presumably contributed by the random high-frequency waves. A soft simple filter - 3 day running average, has been applied to filter out this high-frequency noise (the filter has been chosen in accordance with the observationally known properties of these waves). Power spectrum of the smoothed indices exhibits a stretched exponential decay $E(f) \propto \exp-(f/f_0)^{1/2}$ characteristic to the Hamilton…
There is a long tradition of studying chaotic trajectories in systems whose integrability is broken by means of an external perturbation. Here we explore a different route to chaos, in the dynamics of extended bodies, which arises due to finite-size corrections to the otherwise integrable motion of a test particle. We find that cyclic changes in the overall shape of the body may lead to the onset of chaos. This is applied to the Duffing and Yukawa potentials. For Kepler's potential, periodic deviations from spherical symmetry give rise to chaotic regions around the unperturbed parabolic orbit.
Two subsystems of the Asian Monsoon: the Indian Summer Monsoon and the Western North Pacific Monsoon, have been analysed using their daily indices ISMI and WNPMI. It is shown that on the intraseasonal time scales the ISMI and WNPMI are dominated by the Hamiltonian distributed chaos with the stretched exponential spectrum: $E(f) \propto \exp-(f/f_0)^β$ and analytical values of the parameter $β=3/4$ and $β=1/2$ correspondingly. The relevant daily indices Niño 3 and Niño 4 (with $β=1/2$) of the El Niño-Southern Oscillation (ENSO) and the Australian Monsoon (AUSM index with $β= 1/2$) have been als…
We analyze nonlinear aspects of the self-consistent wave-particle interaction using Hamiltonian dynamics in the single wave model, where the wave is modified due to the particle dynamics. This interaction plays an important role in the emergence of plasma instabilities and turbulence. The simplest case, where one particle (N = 1) is coupled with one wave (M = 1), is completely integrable, and the nonlinear effects reduce to the wave potential pulsating while the particle either remains trapped or circulates forever. On increasing the number of particles (N = 2, M = 1), integrability is lost an…
