Search Results (41)

Classical periodic solution theories of undamped Duffing equations are mostly restricted to continuous nonlinearities, while discontinuous models from mechanical impact and switching circuits lack systematic existence criteria for crossing periodic orbits. This work addresses a second-order undamped Duffing system with jump discontinuity of $g\left(x\right)$ on $x=0$, where $g\left(x\right)$ is piecewise continuously differentiable on two half real axes. With the separation condition ruling out sliding trajectories on the discontinuity line, the continuity of the Poincaré map is proved by Filippov theory for discontinuous ODEs. Using the Poincaré–Bohl fixed-point theorem, we derive multiple sufficient conditions ensuring the existence of $2\pi$ crossing periodic solutions via successive relaxation of growth hypotheses, and a numerical test example confirms the practicability of our theoretical findings, extending classical continuous Duffing results to discontinuous dynamical systems. Full article

This paper studies quantitative forms of approximate recurrence for bounded linear operators on Banach spaces through the notion of uniformly $\delta$-almost periodic vectors. For a prescribed tolerance $\delta \ge 0$, this notion relaxes classical almost periodicity by requiring uniform orbit repetitions up to an error controlled by $\delta$, along relatively dense sets of approximate periods. The main purpose of the paper is to refine this qualitative recurrence condition by introducing return-time profiles. These profiles measure, for each accuracy level, the minimal size of recurrence windows needed to guarantee the existence of an approximate period. Thus, they provide a quantitative refinement of the usual relatively dense return condition. We prove that uniform $\delta$-almost periodicity is equivalent to the finiteness of the associated return-time profile at every positive accuracy level. We also establish basic structural properties of these profiles, including monotonicity with respect to the accuracy and tolerance parameters, behavior under scalar multiplication and forward iteration, and an elementary additive property of approximate periods. The final part of the paper applies the general framework to weighted backward shifts on ${\ell }^p$-spaces. In this setting, the explicit coordinate representation of the iterates allows us to identify several recurrence and obstruction mechanisms. We describe stable threshold recurrence, finite-support recurrence, exact recurrence generated by periodic vectors, and coordinate-level obstructions to $\delta$-almost periodicity. The results provide a rigorous framework for measuring approximate almost periodicity in linear dynamics and clarify how recurrence-window profiles complement the classical qualitative theory of relatively dense returns. Full article

Ion-acoustic waves in dense quantum plasmas are strongly influenced by Fermi degeneracy, Landau quantization, and finite-temperature effects, and in many relevant environments, they also experience memory and nonlocal transport processes that cannot be captured within the planar integer Korteweg-de Vries (KdV) paradigm. In the present work, we revisit this problem by considering a two-fluid, partially degenerate electron-ion plasma in which electron trapping in the presence of a quantizing field and finite temperature is taken into account. Starting from the normalized fluid-Poisson system appropriate for such magnetized quantum plasmas, the reductive perturbation technique is used to derive the planar integer KdV equation for weakly nonlinear ion-acoustic disturbances. Within this integer-order KdV framework, we recast the evolution equation as a planar dynamical system, construct the associated Hamiltonian and effective Sagdeev-like potential, and demonstrate the existence of compressive solitary waves and nonlinear periodic modes via homoclinic and periodic phase-space orbits. Exact multi-soliton solutions and interaction states are then obtained by combining Hirota’s direct bilinear method with generalized Wronskian representations, allowing us to describe not only standard one-, two-, and three-soliton profiles but also positon-negaton interactions relevant to magnetized, partially degenerate plasmas. To incorporate hereditary and history-dependent effects that arise from anomalous transport and nonlocal temporal response in dense environments, we extend the model by introducing a Caputo time-fractional derivative, thereby obtaining a time-fractional KdV (FKdV) equation that continuously connects the classical KdV limit to fractional dynamics. The FKdV equation is analyzed using the Tantawy technique. This semi-analytical iterative scheme yields rapidly convergent series approximations for the fractional ion-acoustic soliton and provides explicit control of the approximation error. The fractional solutions show that varying the order of the Caputo derivative modifies the amplitude, width, and temporal relaxation of the solitary structures and can even split the pulse into two distinct lobes, in contrast with the nearly rigid propagation predicted by the integer-order KdV equation. Taken together, these results clarify how Landau quantization, finite electron temperature, and fractional-order memory jointly shape the morphology, robustness, and interaction properties of ion-acoustic structures in strongly magnetized quantum plasmas of astrophysical and high-energy-density laboratory interest. Full article

Hill’s problem plays an important role in analyzing the local dynamics of an infinitesimal body under the gravitational influence of a distant massive primary and a nearby secondary body of smaller mass. When radiation pressure is included, the resulting model becomes particularly relevant for studying the motion of dust particles and solar-sail spacecraft in the vicinity of minor celestial bodies, such as planets or asteroids. This inclusion breaks the symmetry with respect to the Oy axis that characterizes the configurations of motion in the classical Hill’s problem. Thus, the location of the collinear equilibrium points, and the evolution of the Lyapunov families must be studied independently. Although the planar dynamics of the photogravitational Hill’s problem have been extensively investigated, its three-dimensional structure remains largely unexplored. The present study undertakes a systematic numerical investigation of branches of spatial periodic orbits that bifurcate from the planar Lyapunov families. Specifically, we compute all three-dimensional bifurcations up to multiplicity four and classify them according to their symmetry properties. The analysis reveals that these families exhibit distinct evolutionary patterns in the space of initial conditions, with most of them terminating in collision orbits with the secondary body. Full article

The Collatz map is investigated from a nonlinear-dynamics perspective with emphasis on the structure of its iterative orbits. By embedding integers within Sharkovsky’s ordering, odd initial values are shown to be sufficient for a complete characterization of dynamics. A “direction-phase” decomposition is introduced to separate iterative orbits into upward and downward phases, yielding a family of recursive functions parameterized by the number of upward phases. This formulation reveals a logarithmic scaling relation between the total iteration count and the initial value, confirming finite-time convergence to the period-three orbit. The Collatz dynamics is further shown to be dynamically equivalent to a binary shift map, whose ergodicity implies inevitable evolution toward attractors, thereby reinforcing convergence. Numerical analysis indicates that attraction basins follow a power-law distribution and display pronounced self-similarity. Moreover, odd integers grouped by upward-phase counts are found to follow Gamma statistics. Beyond its research implications, the framework provides a concise pedagogical case study illustrating how nonlinear dynamics, symbolic dynamics, and statistical characterization can be integrated to analyze a classical discrete problem. Full article

Monosynaptic reflexes (MSRs) elicited by constant-intensity group I afferent stimulation exhibit marked amplitude variability, commonly attributed to stochastic presynaptic modulation and dynamic postsynaptic excitability. Here, we tested whether this variability could be attenuated using the Ott–Grebogi–Yorke (OGY) chaos–control algorithm, which stabilizes unstable periodic orbits in low-dimensional nonlinear systems. In spinalized, anesthetized cats, real-time implementation of the OGY method failed to reduce MSR amplitude variability, as quantified by the coefficient of variation, and the return map structure showed no evidence of orbit stabilization. These negative results contrast with successful applications of OGY control in physical systems, cardiac tissue, hippocampal slices, and stochastic neuronal models. We interpret this failure in the context of the intense, ongoing synaptic bombardment characteristic of dorsal horn circuitry, which likely obscures or destroys the low-dimensional geometric structure required for OGY-based control. Our findings delineate a fundamental limit to classical chaos–control algorithms in intact neural circuits and highlight the need for control strategies explicitly robust to high dimensionality and physiological noise. Full article

Digital chaotic systems suffer severe dynamical degradation under finite computational precision, compromising their randomness and unpredictability in security-critical applications. To address this challenge, we introduce the High-Dimensional Delayed Cyclic-Coupled Chaotic Model (HD-DCCCM), a unified framework that integrates high-dimensional coupling, delayed feedback, and time-varying parameter control. In this synergistic design, dynamic perturbations from delays and time-varying signals continuously excite the high-dimensional structure, effectively preventing the collapse into short periodic orbits typical of low-precision environments. Systematic numerical analyses confirm that the HD-DCCCM generates stable hyperchaos with significantly extended periods, consistently outperforming classical maps and representative anti-degradation methods. Moreover, the framework demonstrates strong robustness and flexibility across both homogeneous (identical maps) and heterogeneous (hybrid maps) configurations. These results position the HD-DCCCM as a general and powerful paradigm for constructing degradation-resilient chaotic systems, with broad potential for next-generation secure communications and cryptographic applications. Full article

A classical three-body problem with two planets moving around a central star of variable mass on quasi-periodic orbits is considered. The bodies are assumed to attract each other according to Newton’s law of universal gravitation. The star loses its mass anisotropically, and this leads to the appearance of reactive forces. The problem is analyzed in the framework of Newtonian’s formalism, and equations of motion are derived in terms of the osculating elements of aperiodic motion on quasi-conic sections. As equations of motion are not integrable, the perturbation theory is applied with the perturbing forces expanded into power series in terms of eccentricities and inclinations, which are assumed to be small. Averaging these equations over the mean longitudes of the planets in the absence of mean-motion resonances, we obtain the differential equations describing the long-term evolution of orbital elements. Numerical solutions to the evolution equations are obtained and analyzed for three different three-body systems. The obtained results demonstrate clearly that variability of masses may influence essentially the secular evolution of the orbital elements. All the relevant symbolic and numerical calculations are performed with the computer algebra system Wolfram Mathematica. Full article

Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium—a chaotic billiard—level statistics described by a Brody distribution with parameter $\delta \simeq 0.1$. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio $(\mathrm{PR})\simeq 1150$ compared to the rectangular billiard (regular) case, where the average $\mathrm{PR}\simeq 1500$. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian. Full article

Semiclassical (SC) approximations for various representations of a quantum state are constructed on a single (Lagrangian) surface in the phase space but such surface is not available for chaotic systems. An analogous evolution surface underlies SC representations of the evolution operator, albeit in a doubled phase space. Here, it is shown that corresponding to the Fourier transform on a unitary operator, represented as a Green function or spectral Wigner function, a Legendre transform generates a resolvent surface as the classical basis for SC representations of the resolvent operator in the double-phase space, independently of the integrable or chaotic nature of the system. This surface coincides with derivatives of action functions (or generating functions) depending on the choice of appropriate coordinates, and its growth departs from the energy shell following trajectories in the double-phase space. In an initial study of the resolvent surface based on its caustics, its complex nature is revealed to be analogous to a multidimensional sponge. Resummation of the trace of the resolvent in terms of linear combinations of periodic orbits, known as pseudo orbits or composite orbits, provides a cutoff to the SC sum at the Heisenberg time. Here, it is shown that the corresponding actions for higher times can be approximately included within true secondary periodic orbits, in which heteroclinic orbits join multiple windings of relatively short periodic orbits into larger circuits. Full article

We revisit 77 relaxed (extra)solar multibody (sub)systems containing 2–9 bodies orbiting about gravitationally dominant central bodies. The listings are complete down to (sub)systems with 5 orbiting bodies and additionally contain 33 smaller systems with 2–4 orbiting bodies. Most of the multiplanet systems (68) have been observed outside of our solar system, and very few of them (5) exhibit classical Laplace resonances (LRs). The remaining 9 subsystems have been found in our solar system; they include 7 well-known satellite groups in addition to the four gaseous giant planets and the four terrestrial planets, and they exhibit only one classical Laplace resonant chain, the famous Galilean LR. The orbiting bodies (planets, dwarfs, or satellites) appear to be locked in/near global mean-motion resonances (MMRs), as these are determined in reference to the orbital period of the most massive (most inert) body in each (sub)system. We present a library of these 77 multibody subsystems for future use and reference. The library listings of dynamical properties also include regular spacings of the orbital semimajor axes. Regularities in the spatial configurations of the bodies were determined from patterns that had existed in the mean tidal field that drove multibody migrations toward MMRs, well before the tidal field was erased by the process of `gravitational Landau damping’ which concluded its work when all major bodies had finally settled in/near the global MMRs presently observed. Finally, detailed comparisons of results help us discern the longest commonly-occurring MMR chains, distinguish the most important groups of triple MMRs, and identify a new criterion for the absence of librations in triple MMRs. Full article

The classical many-body problem is not integrable, so perturbation theory based on an exact solution to the two-body problem is usually applied to investigate the dynamics of planetary systems. However, in the case of variable masses, the two-body problem is not integrable, in general, and application of perturbation theory is required to investigate it, as well. In the present paper, we use the perturbation theory to derive the differential equations determining the orbital elements of the relative motion of one body around the other. Two models of the perturbed aperiodic motion on conic and quasi-conic sections are considered and compared. Special attention is paid to the practically important case of small eccentricities, when the perturbing forces may be replaced by the corresponding power series expansions. The differential equations of the perturbed motion are averaged over the mean anomaly, and the evolutionary equations describing the behavior of the orbital elements over long periods of time are obtained for two models. Comparing the corresponding solutions to the evolutionary equations, we have shown that both models demonstrate similar behavior with regard to the secular perturbations of the orbital elements. However, the second model, based on the aperiodic motion on a quasi-conic section, is more appropriate for generalization to the many-body problem with variable masses. All the relevant symbolic and numerical calculations are performed with the computer algebra system Wolfram Mathematica. Full article

A diffractive solar sail is an elegant concept for a propellantless spacecraft propulsion system that uses a large, thin, lightweight surface covered with a metamaterial film to convert solar radiation pressure into a net propulsive acceleration. The latter can be used to perform a typical orbit transfer both in a heliocentric and in a planetocentric mission scenario. In this sense, the diffractive sail, proposed by Swartzlander a few years ago, can be considered a sort of evolution of the more conventional reflective solar sail, which generally uses a metallized film to reflect the incident photons, studied in the scientific literature starting from the pioneering works of Tsander and Tsiolkovsky in the first decades of the last century. In the context of a diffractive sail, the use of a metamaterial film with a Littrow transmission grating allows for the propulsive acceleration magnitude to be reduced to zero (and then, the spacecraft to be inserted in a coasting arc during the transfer) without resorting to a sail attitude that is almost edgewise to the Sun, as in the case of a classical reflective solar sail. The aim of this work is to study the optimal (i.e., the rapid) transfer performance of a spacecraft propelled by a diffractive sail with a Littrow transmission grating (DSLT) in a three-dimensional heliocentric mission scenario, in which the space vehicle transfers between two assigned Keplerian orbits. Accordingly, this paper extends and generalizes the results recently obtained by the author in the context of a simplified, two-dimensional, heliocentric mission scenario. In particular, this work illustrates an analytical model of the thrust vector that can be used to study the performance of a DSLT-based spacecraft in a three-dimensional optimization context. The simplified thrust model is employed to simulate the rapid transfer in a set of heliocentric mission scenarios as a typical interplanetary transfer toward a terrestrial planet and a rendezvous with a periodic comet. Full article

We investigate solutions of the classical Mathieu–Hill (MH) equation that characterizes the dynamics of trapped ions. The analytical model we introduce demonstrates the equations of motion are equivalent to those of a harmonic oscillator (HO). Two independent approaches are used, based on two classes of complex solutions of the MH equation. This paper addresses both a damped HO and parametric oscillator (PO) for an ion confined in an electrodynamic (Paul) trap, along with stability and instability regions for the associated periodic orbits. Full article

Propellantless propulsion systems, such as the well-known photonic solar sails that provide thrust by exploiting the solar radiation pressure, theoretically allow for extremely complex space missions that require a high value of velocity variation to be carried out. Such challenging space missions typically need the application of continuous thrust for a very long period of time, compared to the classic operational life of a space vehicle equipped with a more conventional propulsion system as, for example, an electric thruster. In this context, an interesting application of this propellantless thruster consists of using the solar sail-induced acceleration to artificially precess the apse line of a planetocentric elliptic orbit. This specific mission application was thoroughly investigated about twenty years ago in the context of the GeoSail Technology Reference Study, which analyzed the potential use of a spacecraft equipped with a small solar sail to perform an in situ study of the Earth’s upper magnetosphere. Taking inspiration from the GeoSail concept, this study analyzes the performance of a solar sail-based spacecraft in (artificially) precessing the apse line of a high elliptic orbit around Venus with the aim of exploring the planet’s induced magnetotail. In particular, during flight, the solar sail orientation is assumed to be Sun-facing, and the required thruster’s performance is evaluated as a function of the elliptic orbit’s characteristics by using both a simplified mathematical model of the spacecraft’s planetocentric dynamics and an approximate analytical approach. Numerical results show that a medium–low-performance sail is able to artificially precess the apse line of a Venus-centered orbit in order to ensure the long-term sensing of the planet’s induced magnetotail. Full article