Quasiperiodic Trajectories Research Articles

Stochastic dynamical behaviors of SOS/ERK signaling pathway perturbated by external noise

The SOS/ERK cascades are key signaling pathways that regulate cellular processes ranging from cellular proliferation, differentiation and apoptosis to tumor formation. However, the properties of these signaling pathways are not well understood. More importantly, how stochastic perturbations of internal and external cellular environment affect these pathways remains unanswered. To answer these questions, we, in this paper, propose a stochastic model according to the biochemical reaction processes of the SOS/ERK pathways, and, respectively, research the dynamical behaviors of this model under the four kinds of noises: Gaussian noise, colored noise, Lévy noise and fraction Brown noise. Some important results are found that Gaussian and colored noises have less effect on the stability of the system when the strength of the noise is small; Lévy and fractional Brownian noises significantly change the trajectories of the system. Power spectrum analysis shows that Lévy noise induces a system with quasi-periodic trajectories. Our results not only provide an understanding of the SOS/ERK pathway, but also show generalized rules for stochastic dynamical systems.

A Conservative Memristive Chaotic System with Extreme Multistability and Its Application in Image Encryption.

In this work, a novel conservative memristive chaotic system is constructed based on a smooth memristor. In addition to generating multiple types of quasi-periodic trajectories within a small range of a single parameter, the amplitude of the system can be controlled by changing the initial values. Moreover, the proposed system exhibits nonlinear dynamic characteristics, involving extreme multistability behavior of isomorphic and isomeric attractors. Finally, the proposed system is implemented using STMicroelectronics 32 and applied to image encryption. The excellent encryption performance of the conservative chaotic system is proven by an average correlation coefficient of 0.0083 and an information entropy of 7.9993, which provides a reference for further research on conservative memristive chaotic systems in the field of image encryption.

On the polynomial integrability of the critical systems for optimal eigenvalue gaps

This exploration consists of two parts. First, we will deduce a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent p ∈ (1, ∞) of the Lebesgue spaces concerned. These systems can be used to obtain the optimal lower or upper bounds for eigenvalue gaps of Sturm–Liouville operators and are equivalent to non-convex Hamiltonian systems of two degrees of freedom. Second, with appropriate choices of exponents p, the critical systems are polynomial systems in four dimensions. These systems will be investigated from two aspects. The first one is that by applying the canonical transformation and the Darboux polynomial, we obtain the necessary and sufficient conditions for polynomial integrability of these polynomial critical systems. As a special example, we conclude that the system with p = 2 is polynomial completely integrable in the sense of Liouville. The second is that the linear stability of isolated singular points is characterized. By performing the Poincaré cross section technique, we observe that the systems have very rich dynamical behaviors, including periodic trajectories, quasi-periodic trajectories, and chaos.

Covariance analysis of periodic and quasi-periodic orbits around Phobos with applications to the Martian Moons eXploration mission

Understanding the internal composition of a celestial body is fundamental for formulating theories regarding its origin. Deep knowledge of the distribution of mass under the body’s crust can be achieved by analyzing its moments of inertia and gravity field. In this regard, the two moons of the Martian system have not yet been closely studied and continue to pose questions regarding their origin to the space community; thus, they deserve further characterization. The Martian Moons eXploration mission will be the first of its kind to sample and study Phobos over a prolonged period. This study aims to demonstrate that the adoption of periodic and quasi-periodic retrograde trajectories would be beneficial for the scientific value of the mission. Here, a covariance analysis was implemented to compare the estimation of high-order gravitational field coefficients from different orbital geometries and for different sets of processed observables. It was shown that the adoption of low-altitude non-planar quasi-satellite orbits would help to refine the knowledge of the moon’s libration angle and gravitational field.

Beauty in chaos: the hidden art and sacred geometric designs of trajectories in multi-body gravitational systems

The Earth–Moon gravitational system is naturally chaotic, and mathematical models such as the Circular Restricted Three-Body Problem (CR3BP) employ layers of ordered assumptions in order to enable both trajectory generation and analysis. However, even within this simplified mathematical construct, the underlying chaos of multi-body gravitational systems introduces a pseudo-instability that yield a dynamic and sometimes drastic evolution of trajectory geometry despite seemingly insignificant changes in the initial position and/or velocity of an object. Three-body trajectories feature a natural artistry reminiscent of artistic styles and spiritual motifs originating from diverse time periods and cultures, ranging from sacred geometry and mandalic-like designs indicative of Buddhist and Hindu symbolism, to Spirograph art, and modernist and post-modernist minimalism. This work will analyze the artistic qualities of two classes of three-body trajectories within the Earth–Moon system: (1) cislunar periodic and near-periodic orbits, and (2) the mixture of quasi-periodic and chaotic trajectories of debris particles following a catastrophic breakup event. Overall, this work sets to prove that even in a chaotic multi-body environment, beauty and order emerges through a fine balance of initial conditions.

Sensitivity of different types of structural size combinations to dynamic stability and vibration of angular contact ball bearings based on nonlinear dynamic model

An original nonlinear dynamic model is proposed to study the effects of structural size combinations on the dynamic stability and vibration of ball bearings. The sliding of the ball, whirl motion of cage and vibration of inner ring are studied. The sensitivity of different types of structural size combinations to dynamic stability and vibration is revealed. The results show that the increased ball diameter facilitates the dynamic stability of cage but worsen the vibration of inner ring; the circular, quasi-periodic and chaotic trajectories are induced by the increased outer groove curvature radius, respectively, and the low vibration of inner ring generates at the stable circular trajectories; the large groove bottom circle diameter deteriorates the dynamic stability and vibration.

Computing Natural Transitions Between Tori Near Resonances in the Earth–Moon System

Natural transitions between bounded motions near mean-motion resonances occur throughout our solar system and are valuable in trajectory design. Such phenomena have been examined for natural transitions between periodic orbits near resonances within multi-body systems. However, families of quasi-periodic trajectories, tracing the surface of invariant 2-tori, significantly expand the solution space of bounded motions near resonances. Yet, identifying natural transitions between spatial 2-tori has previously been cumbersome due to the high dimensionality of the associated solution space. This paper approaches the challenge in constructing these natural transfers by using a combination of Poincaré mapping, a well-known technique from dynamical systems theory, and manifold learning, a technique for dimension reduction. The presented approach involves projecting a higher-dimensional dataset of intersections recorded from the hyperbolic invariant manifolds of two 2-tori onto a lower-dimensional embedding, enabling rapid identification of initial guesses for natural transfers. These initial guesses are then corrected and input to a continuation scheme to recover families of geometrically similar transfers connecting families of invariant 2-tori. This approach is demonstrated by constructing families of natural transitions between tori near distinct resonances in the Earth–Moon circular restricted three-body problem.

Periodic and Quasi-Periodic Orbits near Close Planetary Moons

Upcoming missions toward remote planetary moons will fly in chaotic dynamic environments that are significantly perturbed by the oblateness of the host planet. Such a dominant perturbation is often neglected when designing spacecraft trajectories in planetary moon systems. This paper introduces a new time-periodic set of equations of motion that is based on the analytical solution of the zonal equatorial problem and better describes the dynamic evolution of a spacecraft subject to the gravitational attraction of a moon and its oblate host planet. Such a system, hereby referred to as the zonal hill problem, remains populated by resonant periodic orbits and families of two-dimensional quasi-periodic invariant tori that are calculated by means of numerical continuation procedures. The resulting periodic and quasi-periodic trajectories are investigated for the trajectory design of future planetary moons explorers.

E-Z Isomerization in Guanidine: Second-order Saddle Dynamics, Non-statisticality, and Time-frequency Analysis.

Our recent work on the E-Z isomerization reaction of guanidine using ab initio chemical dynamics simulations [Rashmi et al., Regul. Chaotic Dyn. 2021, 26, 119] emphasized the role of second-order saddle (SOS) in the isomerization reaction; however, we could not unequivocally establish the non-statistical nature of the dynamics followed in the reaction. In the present study, we performed thousands of on-the-fly trajectories using forces computed at the MNDO level to investigate the influence of second-order saddle in the E-Z isomerization reaction of guanidine and the role of intramolecular vibrational energy redistribution (IVR) on the reaction dynamics. The simulations reveal that while majority of the trajectories follow the traditional transition state pathways, 15 % of the trajectories follow the SOS path. The dynamics was found to be highly non-statistical with the survival probabilities of the reactants showing large deviations from those obtained within the RRKM assumptions. In addition, a detailed analysis of the dynamics using time-dependent frequencies and the frequency ratio spaces reveal the existence of multiple resonance junctions that indicate the existence of regular dynamics and long-lived quasi-periodic trajectories in the phase space associated with non-RRKM behavior.

Four-body cislunar quasi-periodic orbits and their application to ballistic lunar transfer design

Examining the properties of quasi-periodic orbits provides insight into the Sun-perturbed environment in cislunar space. In this investigation, quasi-periodic trajectories and their properties are explored in the Sun-Earth-Moon four-body problem. Computation and the stability characteristics of families of invariant tori are detailed. Furthermore, this investigation offers a framework for construction of ballistic lunar transfer trajectories in the four-body problem. The framework leverages manifold trajectories to supply a set of initial conditions for construction of periapsis Poincaré maps. Periapsis maps reduce the dimensionality of the space and illuminate solutions of interest as a basis to produce feasible families of transfers. Through a continuation process, families of ballistic transfers and families of transfers that include powered lunar flybys are constructed. Ultimately, these solutions supply an initial guess for transition to the Sun-Earth-Moon ephemeris model.

A Systematic Study about Orbit Flips of Test Particles Caused by Eccentric Von Zeipel–Lidov–Kozai Effects

The problem of orbit flips caused by eccentric von Zeipel–Lidov–Kozai effects is systematically investigated by means of three approaches, including Poincaré sections, dynamical system theory (periodic orbits and invariant manifolds), and perturbation treatments. Poincaré sections show that orbit flips are due to the existence of islands of libration centered at inclination of 90°, dynamical system theory shows that orbit flips are due to the existence of polar periodic orbits and invariant manifolds, and perturbative treatments indicate that orbit flips are due to the libration of a certain critical argument. Using these approaches, the boundaries of flipping regions in the entire parameter space are produced, and they are in excellent agreement with one another. Through analysis, the essence of flipping orbits is reached: (a) flipping orbits are a kind of quasiperiodic trajectory around polar periodic orbits and invariant manifolds at the same level of Hamiltonian provide boundaries of flipping regions, and (b) flipping orbits are a kind of resonant trajectory, and resonant width measures the size of flipping regions.

Exploring the dynamical evolution of minimoon 2020 CD3 in a circular restricted three-body problem framework

Asteroid 2020 CD3 was one of the most recently captured natural objects that became Earth’s transient satellite or minimoon. The asteroid is very likely residing in an Earth-like orbit (ELO) with 1:1 mean-motion resonance and occasionally captured by the Earth. In this study, the dynamical evolution of this object is conducted in a framework of Circular Restricted Three-Body Problem (CR3BP). Five thousand test objects are generated by distributing the orbital space with a random uniform scheme around the object’s nominal values comprising ten sigma-uncertainties of the orbital parameters and four Metonic cycles of the epoch. This numerical task has run for 1000 years with the IAS15 integrator known for its adaptive integration and high accuracy for millions of years of simulation. We found that almost all of the test particles are lying in a boundary region, hence follow chaotic and quasi-periodic trajectories and have become minimoon with an average of 15-16 times. The duration of captures of more than a year is very rare. Interestingly, there seems a preferred recapture time at ∼ 30 years. Most of them still stay in the ELO region, although a few have either been hit the Earth or expelled into vast space.

Multistability and dynamical properties of quantum ion-acoustic flow

Multistability and dynamical properties of ion-acoustic flow are studied in a quantum plasma containing positive beam ions, positive ions and electrons. A four dimensional conservative dynamical system has been proposed for the considered plasma system and is analyzed by considering effects of Mach number and quantum diffraction parameter. Coexistences of multiple chaotic trajectories, chaotic with quasiperiodic and multiperiodic trajectories and chaotic with quasi-periodic and periodic trajectories for ion-acoustic waves are established. The results are suitable for application in comprehending the beam-plasma interaction and studying dynamics of coexisting features in extreme astrophysical plasmas, such as, neutron stars.

Second-order Saddle Dynamics in Isomerization Reaction

The role of second-order saddle in the isomerization dynamics was investigated by considering the $$E-Z$$ isomerization of guanidine. The potential energy profile for the reaction was mapped using the ab initio wavefunction method. The isomerization path involved a torsional motion about the imine (C-N) bond in a clockwise or an anticlockwise fashion resulting in two degenerate transition states corresponding to a barrier of 23.67 kcal/mol. An alternative energetically favorable path ( $$\sim$$ 1 kcal/mol higher than the transition states) by an in-plane motion of the imine (N-H) bond via a second-order saddle point on the potential energy surface was also obtained. The dynamics of the isomerization was investigated by ab initio classical trajectory simulations. The trajectories reveal that isomerization happens via the transition states as well as the second-order saddle. The dynamics was found to be nonstatistical with trajectories exhibiting recrossing and the higher energy second-order saddle pathway preferred over the traditional transition state pathway. Wavelet based time-frequency analysis of internal coordinates indicate regular dynamics and existence of long-lived quasi-periodic trajectories in the phase space.

Large gyro-orbit model of ion velocity distribution in plasma near a wall in a grazing-angle magnetic field

A model is presented for the ion distribution function in a plasma at a solid target with a magnetic field $\boldsymbol {B}$ inclined at a small angle, $\alpha \ll 1$ (in radians), to the target. Adiabatic electrons are assumed, requiring $\alpha \gg \sqrt {Zm_{e}/m_{i}}$ , where $m_{e}$ and $m_{i}$ are the electron and ion mass, respectively, and $Z$ is the charge state of the ion. An electric field $\boldsymbol {E}$ is present to repel electrons, and so the characteristic size of the electrostatic potential $\phi$ is set by the electron temperature $T_{e}$ , $e\phi \sim T_{e}$ , where $e$ is the proton charge. An asymptotic scale separation between the Debye length $\lambda _{D} = \sqrt {\epsilon _0 T_{{e}} / e^{2} n_{{e}} }$ , the ion sound gyro-radius $\rho _{s} = \sqrt { m_{i} ( ZT_{e} + T_{i} ) } / (ZeB)$ and the size of the collisional region $d_{c} = \alpha \lambda _{\textrm {mfp}}$ is assumed, $\lambda _{D} \ll \rho _{s} \ll d_{c}$ . Here $\epsilon _0$ is the permittivity of free space, $n_{e}$ is the electron density, $T_{i}$ is the ion temperature, $B= |\boldsymbol {B}|$ and $\lambda _{\textrm {mfp}}$ is the collisional mean free path of an ion. The form of the ion distribution function is assumed at distances $x$ from the wall such that $\rho _{s} \ll x \ll d_{c}$ , that is, collisions are not treated. A self-consistent solution of the electrostatic potential for $x \sim \rho _{s}$ is required to solve for the quasi-periodic ion trajectories and for the ion distribution function at the target. The large gyro-orbit model presented here allows to bypass the numerical solution of $\phi (x)$ and results in an analytical expression for the ion distribution function at the target. It assumes that $\tau =T_{i}/(ZT_{e})\gg 1$ , and ignores the electric force on the quasi-periodic ion trajectory until close to the target. For $\tau \gtrsim 1$ , the model provides an extremely fast approximation to energy–angle distributions of ions at the target. These can be used to make sputtering predictions.

ENVELOPE FOLLOWING BASED METHOD FOR THE COMPUTATION OF LYAPUNOV EXPONENTS IN ELECTRONIC CIRCUITS

Lyapunov Exponents (LEs) can be considered as a statistical measure giving some approximate but signifficant insight in the behaviour of a dynamic system. If the systems we consider are models of realistic electronic circuits, we can reasonably assume them to be dissipative. LEs are useful to determine some general topological properties of the attractive region so that, even if the state space has a relatively large dimension, it is possible to foresee the presence of stable equilibrium points, periodic or quasi-periodic trajectories or, possibly, chaos. Lyapunov exponents and related measures such as the Lyapunov dimension are used to characterize the dynamics of complex systems and the geometrical properties of their trajectories in the state space. A few methods are available for the numerical computation of these measures and in most cases they are used only for systems that are normalized, well behaved and with a low dimensional state space. It is here proposed an approach that, being based on the envelope following method for the integration of stiff systems, can be efficiently used for the calculation of Lyapunov exponents and dimension in real electronic circuits.

Side-mode suppression in ultraviolet quasi-semicircle microlaser cavity

We report a quasi-semicircle GaN ultraviolet microcavity laser. Poincaré map of the quasi-semicircle cavity shows periodic, quasi-periodic and chaotic optical trajectories. Finite-element-method optical near-field distribution patterns of the periodic optical modes agree with the optical loops in the quasi-semicircle cavity. Based on the numerical results, optically pumped lasing experiment shows that ultraviolet multimode lasing action from different optical resonance modes can be effectively stimulated at low pumping intensity in the quasi-semicircle GaN cavity, and the side-mode suppression phenomenon can be obtained at high pumping intensity. Because the possible period optical loops are quasi-periodic and unstable, the quasi-whispering-gallery mode lasing with higher quality factor prefer to dominate the lasing action under high pumping. The experimental and simulated lasing modes suggest that side-mode suppression phenomenon can be caused by the mode competition between quasi-whispering-gallery modes and other quasi-periodic modes.

Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasiperiodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this "linearization" (or conjugacy) from knowledge of a single quasiperiodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian Problem: determining the characteristics of a quasiperiodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method "the weighted Birkhoff average".

On the landscape of scale invariance in quantum mechanics

We consider the most general scale invariant radial Hamiltonian allowing for anisotropic scaling between space and time. We formulate a renormalisation group analysis of this system and demonstrate numerically the existence of a universal quantum phase transition from a continuous scale invariant phase to a discrete scale invariant phase. Close to the critical point, the discrete scale invariant phase is characterised by an isolated, closed, attracting trajectory in renomalisation group space (a limit cycle). Moving in appropriate directions in the parameter space of couplings this picture is altered to one controlled by a quasi periodic attracting trajectory (a limit torus) or fixed points. We identify a direct relation between the critical point, the renormalisation group picture and the power laws characterising the zero energy wave functions.

Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop

A recent study suggested that the nonlinear feedback loop (NFL) of the three-dimensional nondissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a nontrivial critical point. This study discusses the role of the extension of the NFL in producing quasi-periodic trajectories using a five-dimensional nondissipative Lorenz model (5D-NLM). An analytical solution to the locally linear 5D-NLM is first obtained to illustrate the association of the extended NFL and two incommensurate frequencies whose ratio is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself. While the NFL of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended NFL within the 5D-NLM additionally introduces two new pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. One pair of downscaling and upscaling processes provides a two-way interaction between the original (primary) Fourier modes of the 3D-NLM and the newly-added (secondary) Fourier modes of the 5D-NLM. The other pair of downscaling and upscaling processes involves interactions amongst the secondary modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that the two-way interaction is crucial for producing the quasi-periodic solution. A follow-up study using a 7D nondissipative LM shows that a further extension of NFL, which may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection, is capable of producing one more incommensurate frequency.