Stable Periodic Trajectories Research Articles

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.

Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser

Vector vortex beams are conventionally created as the superposition of orbital angular momentum (OAM) modes with orthogonal polarizations, limiting the available degrees of freedom (DoFs) to 2, while their creation by complex optical devices such as metasurfaces, liquid crystals, and interferometers has hindered their versatility. Here we demonstrate a new class of vector vortex beam constructed from four DoFs as multiple ray-like trajectories with wave-like properties, which we create by operating a simple anisotropic microchip laser in a frequency-degenerate state. Our new structure is obtained by the superposition of two stable periodic ray trajectories, simultaneously fulfilling a completed oscillation in the cavity. By a simple external modulation, we can transform our ray trajectories into vortex beams with large OAM, multiple singularities, as well as exotic helical star-shaped patterns. Our experimental results are complemented by a complete theoretical framework for this new class of beam, revealing parallels to hybrid SU(2) coherent states. Our approach offers in principle unlimited DoFs for vectorial structured light with concomitant applications, for example, in engineering classically entangled light and in vectorial optical trapping and tweezing.

A non-local reduction principle for cocycles in Hilbert spaces

We study cocycles (non-autonomous dynamical systems) satisfying a certain squeezing condition with respect to the quadratic form of a bounded self-adjoint operator acting in a Hilbert space. We prove that (under additional assumptions) the orthogonal projector maps the fibres of some invariant set, containing bounded trajectories, in a one-to-one manner onto the negative subspace of the operator. This allows to reduce interesting dynamics onto this invariant set, which in some cases can be considered as a kind of inertial manifold for the cocycle. We consider applications of the reduction principle for periodic cocycles. For such cocycles we give an extension of the Massera second theorem, obtain the conditions for the existence of a Lyapunov stable periodic trajectory and prove convergence-type results, which we apply to study nonlinear periodic in time delayed-feedback equations posed in a proper Hilbert space and parabolic problems with a nonlinear periodic in time boundary control. The required operator is obtained as a solution to certain operator inequalities with the use of Yakubovich-Likhtarnikov frequency theorem for C0-semigroups and its properties are established from the Lyapunov inequality and dichotomy of the linear part of the problem.

Conservative, dissipative and super-diffusive behavior of a particle propelled in a regular flow

A recent model of Ariel et al. (2017) for explaining the observation of Lévy walks in swarming bacteria suggests that self-propelled, elongated particles in a periodic array of regular vortices perform a super-diffusion that is consistent with Lévy walks. The equations of motion, which are reversible in time but not volume preserving, demonstrate a new route to Lévy walking in chaotic systems. Here, the dynamics of the model is studied both analytically and numerically. It is shown that the apparent super-diffusion is due to “sticking” of trajectories to elliptic islands, regions of quasi-periodic orbits reminiscent of those seen in conservative systems. However, for certain parameter values, these islands coexist with asymptotically stable periodic trajectories, causing dissipative behavior on very long time scales.

Discontinuity-induced intermittent synchronization transitions in coupled non-smooth systems.

The synchronization transition in coupled non-smooth systems is studied for increasing coupling strength. The average order parameter is calculated to diagnose synchronization of coupled non-smooth systems. It is found that the coupled non-smooth system exhibits an intermittent synchronization transition from the cluster synchronization state to the complete synchronization state, depending on the coupling strength and initial conditions. Detailed numerical analyses reveal that the discontinuity always plays an important role in the synchronization transition of the coupled non-smooth system. In addition, it is found that increasing the coupling strength leads to the coexistence of periodic cluster states. Detailed research illustrates that the periodic clusters consist of two or more coexisting periodic attractors. Their periodic trajectory passes from one region to another region that is divided by discontinuous boundaries in the phase space. The mutual interactions of the local nonlinearity and the spatial coupling ultimately result in a stable periodic trajectory.

Nonlinear dynamics of displaced non-Keplerian orbits with low-thrust propulsion

This paper discusses the stability, transition and control of displaced non-Keplerian orbits by the spacecraft using low-thrust propulsion. The two-body dynamical model developed in the polar coordinates is parameterized by the thrust pitch angle, and then two of the hyperbolic and elliptic equilibria are solved from it. The bounded motions near two equilibria are investigated by dynamical system techniques to find out all the stable and unstable periodic trajectories, and two scenarios of the resonant periodic trajectory are presented. Regardless of the thrust pitch angle, all the transit orbits are numerically demonstrated to be restricted inside the invariant manifolds of Lyapunov orbit near the hyperbolic equilibrium. Then the transit orbits can be distinguished from non-transit ones by the restriction of three-dimensional invariant manifolds projected onto the Poincaré section or position space. Based on the influence of thrust direction on the system topology, operating the thrust pitch angle is an effective tool to achieve the transfer within different types of KAM tori, or even transfer beyond the KAM tori.

Almost global asymptotic stability of a grid-connected synchronous generator

We study the global asymptotic behavior of a grid-connected constant field current synchronous generator (SG). The grid is regarded as an “infinite bus,” i.e., a three-phase AC voltage source. The generator does not include any controller other than the frequency droop loop. This means that the mechanical torque applied to this generator is an affine function of its angular velocity. The negative slope of this function is the frequency droop constant. We derive sufficient conditions on the SG parameters under which there exist exactly two periodic state trajectories for the SG, one stable and another unstable, and for almost all initial states, the state trajectory of the SG converges to the stable periodic trajectory (all the angles are measured modulo $$2\pi $$ ). Along both periodic state trajectories, the angular velocity of the SG is equal to the grid frequency. Our sufficient conditions are easy to check computationally. An important tool in our analysis is an integro-differential equation called the exact swing equation, which resembles a forced pendulum equation and is equivalent to our fourth-order model of the grid-connected SG. Apart from our objective of providing an analytical proof for a global asymptotic behavior observed in a classical dynamical system, a key motivation for this work is the development of synchronverters which are inverters that mimic the behavior of SGs. Understanding the global dynamics of SGs can guide the choice of synchronverter parameters and operation. As an application we find a set of stable nominal parameters for a 500-kW synchronverter.

Graph-manifolds and integrable Hamiltonian systems

We study the topology of the three-dimensional constant- energy manifolds of integrable Hamiltonian systems realizable in the form of a special class of so-called `molecules'. Namely, for this class of manifolds the Reidemeister torsion is calculated in terms of the Fomenko-Zieschang invariants. A connection between the torsion of a constant-energy manifold and stable periodic trajectories is found. Bibliography: 17 titles.

The Coexistence of Invariant Tori and Topological Horseshoe in a Generalized Nosé–Hoover Oscillator

This paper is devoted to the study of dynamical complexity of a generalized Nosé–Hoover oscillator which is a three-dimensional quadratic polynomial system. Precisely, a lot of moderately conservative regions are found, each of which is filled with different sequences of nested tori with various knot types and is embedded in the “chaotic region”. This shows that the generalized Nosé–Hoover oscillator may possess so-called “fat fractal” structure in phase space. In addition, horseshoe chaos can be demonstrated by applying the topological horseshoe theory to a Poincaré map defined in a proper cross-section, which further shows the coexistence of infinitely stable periodic trajectories and infinite saddle periodic trajectories.

Modeling and dynamics of a two-line kite

• Lagrangian model for a two-line kite. • Analysis of natural symmetric and asymmetric equilibrium states. • Floquet analysis of natural and driven periodic orbits. • Chaotic dynamics for periodically driven kites. • High tensions and cross-wind conditions without control are shown. A mathematical model of a kite connected to the ground by two straight tethers of varying lengths is presented and used to study the traction force generated by kites flying in cross-wind conditions. The equations of motion are obtained by using a Lagrangian formulation, which yields a low-order system of ordinary differential equations free of constraint forces. Two parameters are chosen for the analysis. The first parameter is the wind velocity. The second parameter is one of the stability derivatives of the aerodynamic model: the roll response to the sideslip angle, known also as effective dihedral. This parameter affects significantly the lateral dynamics of the kite. It has been found that when the effective dihedral is below a certain threshold, the kite follows stable periodic trajectories, and naturally flies in cross-wind conditions while generating a high tension along both tethers. This result indicates that kite-based propulsion systems could operate without controlling tether lengths if kite design, including the dihedral and sweep angles, is done appropriately. If both tether lengths are varied out-of-phase and periodically, then kite dynamics can be very complex. The trajectories are chaotic and intermittent for values of the effective dihedral below a certain negative threshold. It is found that tether tensions can be very similar with and without tether length modulation if the parameters of the model are well-chosen. The use of the model for pure traction applications of kites is discussed.

Dynamic Characteristics of Periodic Motions in a Restricted N-Body Problem

AbstractWe investigate the existence and follow the evolution and dynamic stability of periodic motions in a restricted version of the N-body problem, where the positions of the mass bodies reside persistently on a planar polygon ring and its geometrical centre. The results show the existence of both (linearly) stable and unstable periodic orbits around the body placed on the ring centre (inside the ring), around all the periphery bodies (outside the ring) and around one or more of the periphery bodies. The stable periodic trajectories form regions of gravitational stability, whereas some classes of the unstable orbits exhibit bifurcations to non-symmetric periodic orbits of the same period, and less often period-doubling bifurcations.

BIFURCATION CONTROL OF A PREDATOR–PREY MODEL BASED ON NUTRITION KINETICS

The existence conditions of Hopf bifurcation for a predator–prey model based on nutrition kinetics are given. The two results may appear as follows: one is that the model has a stable periodic trajectory from Hopf bifurcation, which shows the system is in an ecological balance; the other is that periodic trajectory from Hopf bifurcation is unstable, which indicates the system is in a sharp or catastrophic loss of stability. For the latter, a bifurcation controller is designed to make the periodic trajectory stable. Finally, some simulations are carried out to prove the results.

Topological degree in analysis of canard-type trajectories in 3-D systems

Piecewise linear systems become increasingly important across a wide range of engineering applications spurring an interest in developing new mathematical models and methods of their analysis, or adapting methods of the theory of smooth dynamical systems. One such areas is the design of controllers which support the regimes of operation described by canard trajectories of the model, including applications to engineering chemical processes, flight control, electrical circuits design, and neural networks. In this article, we present a scenario which ensures the existence of a topologically stable periodic (cyclic) canard trajectory in slow-fast systems with a piecewise linear fast component. In order to reveal the geometrical structure responsible for the existence of the canard trajectory, we focus on a simple prototype piecewise linear nonlinearity. The analysis is based on application of the topological degree.

Periodic Orbits and Equilibria in Glass Models for Gene Regulatory Networks

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Glass models are frequently used to model gene regulatory networks. A distinct feature of the Glass model is that its dynamics can be formalized as paths through multi-dimensional binary hypercubes. In this paper, we report a broad range of results about Glass models that have been obtained by computing the binary codes that correspond to the hypercube paths. Specifically, we propose algorithmic methods for the synthesis of specific Glass networks based on these codes. In contrast to existing work, bi-periodic networks and networks possessing both stable equilibria and periodic trajectories are considered. The robustness of the attractor is also addressed, which gives rise to hypercube paths with nondominated nodes and double coils. These paths correspond to novel combinatorial problems, for which initial experimental results are presented. Finally, a classification of Glass networks with respect to their corresponding gene interaction graphs for three genes is presented. </para>

On the stability of periodic trajectories in some gene network models

Under study is the stability of the periodic trajectories of the gene network models with negative feedback. Some sufficient conditions are given for the existence of stable periodic trajectories.

Fractional standard map

Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the map parameter and the fractional order of the derivative in the original differential equation, this nonlinear dynamical system demonstrates attractors (fixed points, stable periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-like structures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors in the chaotic sea was observed.

On Robust Stability And Sensitivity Of Circadian Rhythms

We develop an approach to robustness analysis of circadian rhythms from the viewpoints of (local) robust stability of periodic trajectories and their period-sensitivity, based on mathematical models. Our attention is directed to the core molecular model generating circadian rhythms in Drosophila, proposed by Goldbeter [1], for our approach to be illustrated. We consider two cases when the circadian rhythms are subjected to periodic forcing by light/dark (LD) cycles and when the circadian rhythms are under continuous darkness (DD). First, we propose conditions/algorithms for (local) robust stability, and show that the circadian rhythms generated by the core molecular model are (locally) robustly stable, in both LD and DD cases. Second, in DD case, we derive a formula of period sensitivity, and assess effects of (infinitesimal) parameter-variations on the period of the circadian rhythms.

Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems

We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedomcorresponding to fast motion and the other one corresponding to slow motion. The Hamiltonianfunction is the sum of potential and kinetic energies, the kinetic energy being a weighted sumof squared momenta. The ratio of time derivatives of slow and fast variables is of order$\epsilon $«$ 1$. At frozen values of the slow variables there is a separatrix on the phaseplane of the fast variables and there is a region in the phase space (the domain ofseparatrix crossings) where the projections of phase points onto the plane of the fastvariables repeatedly cross the separatrix in the process of evolution of the slowvariables. Under a certain symmetry condition we prove the existence of many, of order$1/\epsilon$, stable periodic trajectories in the domain of the separatrix crossings. Each ofthese trajectories is surrounded by a stability island whose measure is estimated frombelow by a value of order $\epsilon$. Thus, the total measure of the stability islands isestimated from below by a value independent of $\epsilon$. We find the location of stableperiodic trajectories and an asymptotic formula for the number of these trajectories. Asan example, we consider the problem of motion of a charged particle in the parabolicmodel of magnetic field in the Earth magnetotail.

Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems

We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter ɛ of the problem. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ɛ) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ɛ. So, the total measure of the stability islands is estimated from below by a value independent of ɛ. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map.

On the absence of stable periodic orbits in domains of separatrix crossings in nonsymmetric slow-fast Hamiltonian systems

We consider a two degree of freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. We assume that at frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under rather general conditions, we prove that there are no stable periodic trajectories of any prescribed period inside the domain of separatrix crossings, except maybe for periodic trajectories passing anomalously close to the saddle point.