Narrow Results

The regularized long-wave ($ℝ𝕃𝕎$) equation is a fundamental model in shallow water wave theory, with extended relevance to other physical systems, such as plasma physics. The $ℝ𝕃𝕎$ equation governs key nonlinear wave phenomena, including the propagation and interaction of solitons, or localized solitary waves. In plasma physics, it describes ion-acoustic waves, which are low-frequency compressional waves arising from the interaction between ions and electrons. While the ion-acoustic wave exhibits Korteweg–de Vries soliton behavior in the cold plasma limit, the $ℝ𝕃𝕎$ equation offers a more accurate representation of ion-acoustic solitons and peaked structures in warm plasmas by accounting for thermal effects.

This research presents novel analytical approximations for solitary wave solutions within the $ℝ𝕃𝕎$ equation, which is critical for modeling nonlinear shallow water and plasma waves. The equation’s ability to describe peaked solitary waves, known as “peakons”, represents wave-breaking phenomena. By employing the improved Riccati expansion and modified Fan expansion techniques, this study derives periodic peakon solutions, offering new insights into the equation’s behavior.

A thorough analysis of the $ℝ𝕃𝕎$ equation is provided, emphasizing its importance for nonlinear wave modeling in both fluid and plasma systems. This paper also includes numerical validation using He’s variational iteration method, which enhances the transparency of the findings by addressing the underlying assumptions and limitations. The principal findings contribute to the broader understanding of nonlinear solitary wave theory, with practical implications for nonlinear wave dynamics across multiple physical domains. Suggested extensions are outlined for further investigation within this established theoretical framework. This study maintains consistent notation and terminology to facilitate clear communication of ideas in adherence to academic standards.

Let $M$ be a $2n$-dimensional ($n\ge 2$), compact smooth Riemannian manifold endowed with a symplectic form $\omega$. In this paper, we show that, if a symplectic diffeomorphism $f$ is $C^1$-robustly measure expansive, then it is Anosov and a $C^1$ generic measure expansive symplectic diffeomorphism $f$ is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system $(H,e,{{ℰ}}_{H,e})$ is robustly measure expansive, then $(H,e,{{ℰ}}_{H,e})$ is Anosov.

In this paper we introduce a symplectic explicit RKN method for Hamiltonian systems with periodical solutions. The method has algebraic order four and phase-lag order six at a cost of four function evaluations per step. Numerical experiments show the relevance of the developed algorithm. It is found that the new method is much more efficient than the standard symplectic fourth-order method.

By comparing with symplectic different methods, the quadratic element is an approximately symplectic method which can keep high accuracy approximate of symplectic structure for Hamiltonian chaos, and it is also energy conservative when there have chaos phenomenon. We use the quadratic finite element method to solve the H$ê$non–Heiles system, and this method was never used before. Combining with Poincar$ê$ section, when we increase the energy of the systems, KAM tori are broken and the motion from regular to chaotic. Without chaos, three kinds of methods to calculate the Poincar$ê$ section point numbers are the same, and the numbers are different with chaos. In long-term calculation, the finite element method can better keep dynamic characteristics of conservative system with chaotic motion.

In this paper, the perturbation theory is extended to be applicable for systems containing conformable derivative of fractional order $\alpha$. This is needed as an essential and powerful approximation method for describing systems with conformable differential equations that are difficult to solve analytically. The work here is derived and discussed for the conformable Hamiltonian systems that appears in the conformable quantum mechanics. The required $\alpha$-corrections for the energy eigenvalues and eigenfunctions are derived. To demonstrate this extension, three illustrative examples are given, and the standard values obtained by the traditional theory are recovered when $\alpha =1$.

We apply the Batalin–Fradkin–Fradkina–Tyutin formalism to a prototypical second-class system, aiming to convert its constraints from second class to first class. The proposed system admits a consistent initial set of second-class constraints and an open potential function providing room for feasible applications to field theory and mechanical models. The constraints can be arbitrarily nonlinear, broadly generalizing previously known cases. We obtain a sufficient condition for which a simple closed expression for the Abelian converted constraints and modified involutive Hamiltonian can be achieved. As an explicit example, we discuss a spontaneous Lorentz symmetry breaking vectorial model, obtaining the full first-class Abelianized constraints in closed form and the corresponding involutive Hamiltonian.

This contribution summarizes work on finite, non-cyclic Hamiltonian systems —in particular the one-dimensional finite oscillator—, in conjunction with a Lie algebraic definition of the (meta-) phase space of finite systems, and a corresponding Wigner distribution function for the state vectors. The consistency of this approach is important for the strategy of fractionalization of a finite Fourier transform, and the contraction of finite unitary to continuous symplectic transformations of Hamiltonian systems.

We present preliminary numerical findings concerning measure synchronization in a pair of coupled Nonlinear Hamiltonian Systems (NLHS) derived from a Nonlinear Schrödinger Equation (NLSE). The dynamics of the two coupled NLHS were found to exhibit a transition to coherent invariant measure; their orbits sharing the same phase space as the coupling strength is increased. Transitions from quasiperiodicity (QP) measure desynchronization to QP measure synchronization and QP measure desynchronization to chaotic (CH) measure synchronization were observed.

Partial synchronization in Hamiltonian systems is investigated based on the concept of measure-synchronization. The classical φ⁴ model is used for the investigation. A macroscopic observable of long-term average of particle energy is computed to describe transitions between desynchronization, different partial synchronization, and complete synchronization structures. It is found that, prior to the entire synchronization of all oscillators, partial measure-synchronization for some clusters of oscillators is stable within certain regions. Moreover, transition from quasiperiodicity to chaos is observed to be associated with the measure-synchronization as the coupling strength is increased.

At a saddle-center bifurcation for Hamiltonian systems, the homoclinic orbit is cusp shaped at the nonlinear nonhyperbolic saddle point. Near but before the bifurcation, orbits are periodic corresponding to the unfolding of the homoclinic orbit, while after the bifurcation a double homoclinic orbit is formed with a local and global branch. The saddle-center bifurcation is dynamically unfolded due to a slowly varying potential. Near the unfolding of the homoclinic orbit, the period and action are analyzed. Asymptotic expansions for the action, period and dissipation are obtained in an overlap region near the homoclinic orbit of the saddle-center bifurcation. In addition, the unfoldings of the action and dissipation functions associated with zero energy orbits (periodic and homoclinic) near the saddle-center bifurcation are determined using the method of matched asymptotic expansions for integrals.

The actuator saturation effects on the global state space structure of a rotating arm with a Hamiltonian control law are studied via bifurcation analysis. In addition to other well-known codimension-two bifurcation points, some saturation-induced "fan-like" bifurcation points are detected. The dynamical complexity found in this seemingly simple example is remarkable.

We employ Chirikov's overlap criterion to investigate interactions between subharmonic resonances of coherent structure solutions of the Gross–Pitaveskii (GP) equation governing the mean-field dynamics of cigar-shaped Bose–Einstein condensates in optical superlattices. We apply a standing wave ansatz to the GP equation to obtain a parametrically forced Duffing equation describing the BEC's spatial dynamics. We then investigate analytically the dependence of spatial resonances on the depth of the superlattice potential, deriving an order-of-magnitude estimate for the critical depth at which spatial resonances with respect to different lattice harmonics first overlap. We also derive a formula for the size of resonance zones and examine changes in our estimates as the relative superlattice amplitudes corresponding to the different harmonics are adjusted. We investigate the onset of global chaos and support our analytical work with numerical simulations.

We investigate the connection between local and global dynamics of two N-degree of freedom Hamiltonian systems with different origins describing one-dimensional nonlinear lattices: The Fermi–Pasta–Ulam (FPU) model and a discretized version of the nonlinear Schrödinger equation related to Bose–Einstein Condensation (BEC). We study solutions starting in the vicinity of simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phase motion (OPM), which are known in closed form and whose linear stability can be analyzed exactly. Our results verify that as the energy E increases for fixed N, beyond the destabilization threshold of these orbits, all positive Lyapunov exponents L_i, i = 1,…, N - 1, exhibit a transition between two power laws, L_i ∝ E^B_k, B_k > 0, k = 1, 2, occurring at the same value of E. The destabilization energy E_c per particle goes to zero as N → ∞ following a simple power-law, E_c/N ∝ N^-α, with α being 1 or 2 for the cases we studied. However, using SALI, a very efficient indicator we have recently introduced for distinguishing order from chaos, we find that the two Hamiltonians have very different dynamics near their stable SPOs: For example, in the case of the FPU system, as the energy increases for fixed N, the islands of stability around the OPM decrease in size, the orbit destabilizes through period-doubling bifurcation and its eigenvalues move steadily away from -1, while for the BEC model the OPM has islands around it which grow in size before it bifurcates through symmetry breaking, while its real eigenvalues return to +1 at very high energies. Furthermore, the IPM orbit of the BEC Hamiltonian never destabilizes, having finite-size islands around it, even for very high N and E. Still, when calculating Lyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunov exponents decrease following an exponential law and yield extensive Kolmogorov–Sinai entropies per particle h_KS/N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

In this paper, we consider compact, invariant sets in Hamiltonian systems in order to extend the concept of crisis to such systems. We focus on crisis-induced intermittency in several systems where two invariant sets merge, obtaining scaling laws for the residence times and for the probability distribution decay as a function of a critical parameter. The connection to hitherto known crisis-induced intermittency in dissipative systems is discussed.

In this paper, we study the problem of transition from integrable (regular) to nonintegrable (chaotic) dynamics in a family of Hamiltonian systems . We show that this transition happens exactly at bifurcation of the periodic orbit of the Hamiltonian system, when its stability discriminant Δ(ε) = +2. As an example, we investigate the system .

In the present tutorial we address a problem with a long history, which remains of great interest to date due to its many important applications: It concerns the existence and stability of periodic and quasiperiodic orbits in N-degree of freedom Hamiltonian systems and their connection with discrete symmetries. Of primary importance in our study is what we call nonlinear normal modes (NNMs), i.e. periodic solutions which represent continuations of the system's linear normal modes in the nonlinear regime. We examine questions concerning the existence of such solutions and discuss different methods for constructing them and studying their stability under fixed and periodic boundary conditions.

In the periodic case, we find it particularly useful to approach the problem through the discrete symmetries of many models, employing group theoretical concepts to identify a special type of NNMs which we call one-dimensional "bushes". We then describe how to use linear combinations of s ≥ 2 such NNMs to construct s-dimensional bushes of quasiperiodic orbits, for a wide variety of Hamiltonian systems including particle chains, a square molecule and octahedral crystals in 1, 2 and 3 dimensions. Next, we exploit the symmetries of the linearized equations of motion about these bushes to demonstrate how they may be simplified to study the destabilization of these orbits, as a result of their interaction with NNMs not belonging to the same bush. Applying this theory to the famous Fermi Pasta Ulam (FPU) chain, we review a number of interesting results concerning the stability of NNMs and higher-dimensional bushes, which have appeared in the recent literature.

We then turn to a newly developed approach to the analytical and numerical construction of quasiperiodic orbits, which does not depend on the symmetries or boundary conditions of our system. Using this approach, we demonstrate that the well-known "paradox" of FPU recurrences may in fact be explained in terms of the exponential localization of the energies E_q of NNM's being excited at the low part of the frequency spectrum, i.e. q = 1, 2, 3, …. These results indicate that it is the stability of these low-dimensional compact manifolds called q-tori, that is related to the persistence or FPU recurrences at low energies. Finally, we discuss a novel approach to the stability of orbits of conservative systems, expressed by a spectrum of indices called GALI_k, k = 2, …, 2N, by means of which one can determine accurately and efficiently the destabilization of q-tori, leading, after very long times, to the breakdown of recurrences and, ultimately, to the equipartition of energy, at high enough values of the total energy E.

We review the dynamical mechanisms we have found to support the morphological features in barred-spiral galaxies based on chaotic motions of stars in their gravitational fields. These morphological features are the spiral arms, that emerge out of the ends of the bar, but also shape the bar itself. The potentials used have been estimated directly from near-infrared images of barred-spiral galaxies. In this paper, we present the results from the study of the dynamics of the potentials of the galaxies NGC 4314, NGC 1300 and NGC 3359. The main unknown parameter in our models is the pattern speed of the system Ω_p. By varying Ω_p, we have investigated several cases trying to match the results of our modeling with available photometrical and kinematical data. We found realistic models with stars on spirals in chaotic motion, while their bars are built by stars usually on regular orbits. However, we also encountered cases, where a major part of trajectories of the stars even in the bar is chaotic as well. Finally, we examined the gas dynamics of barred-spiral systems, and found that the presence of gas reinforces the intensity of the "chaotic" spiral arms.

The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of [Maffione et al., 2011b] for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed.

This work deals with both problems. We first extend the work of [Maffione et al., 2011b] for mappings to the 2D [Hénon & Heiles, 1964] potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectra of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by [Jorba & Zou, 2005] (called taylor), and we compare its performance with other two well-known efficient integrators: the [Prince & Dormand, 1981] implementation of a Runge–Kutta of order 7–8 (DOPRI8) and a Bulirsch–Stöer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator.

We first show that a combination of the FLI/OFLI, the MEGNO and the GALI_2N succeeds in describing in detail most of the dynamical characteristics of a general Hamiltonian system. In the second part, we show that the precision of taylor is better than that of the other integrators tested, but it is not well suited to integrate systems of equations which include the variational ones, like in the computing of almost all the preceeding indicators of chaos. The result which induces us to draw this conclusion is that the computing times spent by taylor are far greater than the times consumed by the DOPRI8 and the Bulirsch–Stöer integrators in such cases. On the other hand, the package is very efficient when we only need to integrate the equations of motion (both in precision and speed), for instance to study the chaotic diffusion. We also notice that taylor attains a greater precision on the coordinates than either the DOPRI8 or the Bulirsch–Stöer.

We study the problem of efficient integration of variational equations in multidimensional Hamiltonian systems. For this purpose, we consider a Runge–Kutta-type integrator, a Taylor series expansion method and the so-called "Tangent Map" (TM) technique based on symplectic integration schemes, and apply them to the Fermi–Pasta–Ulam β (FPU-β) lattice of N nonlinearly coupled oscillators, with N ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique — which shows the best performance among the tested algorithms — and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.

In this paper, we study the topology of isochronous centers of Hamiltonian differential systems with polynomial Hamiltonian functions $H(x,y)$ such that the isochronous center lies on the level curve $H(x,y)=0$. We prove that, in the one-dimensional homology group of the Riemann surface (removing the points at infinity) of level curve $H(x,y)=h$, the vanishing cycle of an isochronous center cannot belong to a subgroup generated by those small loops such that each of them is centered at a removed point at infinity of having one of the two special types described in the paper, where $h$ is sufficiently close to $0$. Besides, we present some topological properties of isochronous centers for a large class of Hamiltonian systems of degree $n$, whose homogeneous parts of degree $n$ contain factors with multiplicity of no more than $n/2$. As applications, we study the nonisochronicity for some Hamiltonian systems with quite complicated forms which are usually very hard to handle by the classical tools.