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 and chaos develops. Our analyses identify the two standard ways for chaos to appear and grow (the homoclinic tangle born from a separatrix, and the resonance overlap near an elliptic fixed point). Moreover, a strong form of chaos occurs when the energy is high enough for the wave amplitude to vanish occasionally.
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…
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）
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.
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…
We provide here a comprehensive proof that the so-called Labyrinth chaos systems, a member of the Thomas-Rössler (TR) class of systems do not admit a Hamiltonian; yet they admit a vector potential. The proof starts from the general case of TR systems, which are in general non-conservative and we show that this is also true for the conservative (volume-preserving) case known as `Labyrinth chaos'. To our knowledge, this is the first instance reported where a conservative chaotic system does not, in principle, admit a Hamiltonian symplectic structure. Still, a vector potential is readily admissib…
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.
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…
By identifying Hamiltonian flows with geodesic flows of suitably chosen Riemannian manifolds, it is possible to explain the origin of chaos in classical Newtonian dynamics and to quantify its strength. There are several possibilities to geometrize Newtonian dynamics under the action of conservative potentials and the hitherto investigated ones provide consistent results. However, it has been recently argued that endowing configuration space with the Jacobi metric is inappropriate to consistently describe the stability/instability properties of Newtonian dynamics because of the non-affine param…
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。
The competition between strength and correlation of coupling terms in a Hamiltonian defines numerous phenomenological models exhibiting spectral properties interpolating between those of Poisson (integrable) and Wigner-Dyson (chaotic) ensembles. It is important to understand how the off-diagonal terms of a Hamiltonian evolve as one or more symmetries of an integrable system are explicitly broken. We introduce a deformed Poisson ensemble to demonstrate an exact mapping of the coupling terms to the underlying symmetries of a Hamiltonian. From the maximum entropy principle we predict a chaotic li…
We study the network replicator equation and characterize its fixed points on arbitrary graph structures for $2 \times 2$ symmetric games. We show a relationship between the asymptotic behavior of the network replicator and the existence of an independent vertex set in the graph and also show that complex behavior cannot emerge in $2 \times 2$ games. This links a property of the dynamical system with a combinatorial graph property. We contrast this by showing that ordinary rock-paper-scissors (RPS) exhibits chaos on the 3-cycle and that on general graphs with $\geq 3$ vertices the network repl…
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 …
We show that the Ulam method applied to dynamical symplectic maps generates Ulam networks which belong to the class of small world networks appearing for social networks of people, actors, power grids, biological networks and Facebook. We analyze the small world properties of Ulam networks on examples of the Chirikov standard map and the Arnold cat map showing that the number of degrees of separation, or the Erdös number, grows logarithmically with the network size for the regime of strong chaos. This growth is related to the Lyapunov instability of chaotic dynamics. The presence of stability …
Drastic facilitation of the onset of global chaos in a periodically driven Hamiltonian system due to an extremum in the dependence of eigenfrequency on energy
