Showing papers in "Regular & Chaotic Dynamics in 2020"

Highly Dispersive Optical Solitons of an Equation with Arbitrary Refractive Index

Abstract: A nonlinear fourth-order differential equation with arbitrary refractive index for description of the pulse propagation in an optical fiber is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we look for solutions of the equation using the traveling wave reduction. We present a novel method for finding soliton solutions of nonlinear evolution equations. The essence of this method is based on the hypothesis about the possible type of an auxiliary equation with an already known solution. This new auxiliary equation is used as a basic equation to look for soliton solutions of the original equation. We have found three forms of soliton solutions of the equation at some constraints on parameters of the equation.

A New $$(3+1)$$ -dimensional Hirota Bilinear Equation: Its Bäcklund Transformation and Rational-type Solutions

Abstract: The behavior of specific dispersive waves in a new $$(3+1)$$ -dimensional Hirota bilinear (3D-HB) equation is studied. A Backlund transformation and a Hirota bilinear form of the model are first extracted from the truncated Painleve expansion. Through a series of mathematical analyses, it is then revealed that the new 3D-HB equation possesses a series of rational-type solutions. The interaction of lump-type and 1-soliton solutions is studied and some interesting and useful results are presented.

Conservation Laws for Highly Dispersive Optical Solitons in Birefringent Fibers

Abstract: This paper reports conservation laws for highly dispersive optical solitons in birefringent fibers. Three forms of nonlinearities are studied which are Kerr, polynomial and nonlocal laws. Power, linear momentum and Hamiltonian are conserved for these types of nonlinear refractive index.

A Map for Systems with Resonant Trappings and Scatterings

Abstract: Slow-fast dynamics and resonant phenomena can be found in a wide range of physical systems, including problems of celestial mechanics, fluid mechanics, and charged particle dynamics. Important resonant effects that control transport in the phase space in such systems are resonant scatterings and trappings. For systems with weak diffusive scatterings the transport properties can be described with the Chirikov standard map, and the map parameters control the transition between stochastic and regular dynamics. In this paper we put forward the map for resonant systems with strong scatterings that result in nondiffusive drift in the phase space, and trappings that produce fast jumps in the phase space. We demonstrate that this map describes the transition between stochastic and regular dynamics and find the critical parameter values for this transition.

Optical Dromions and Domain Walls with the Kundu - Mukherjee - Naskar Equation by the Laplace - Adomian Decomposition Scheme

Abstract: This paper numerically addresses optical dromions and domain walls that are monitored by Kundu – Mukherjee – Naskar equation. The Kundu – Mukherjee – Naskar equation is considered because this model describes the propagation of soliton dynamics in optical fiber communication system. The scheme employed in this work is Laplace – Adomian decomposition type. The accuracy of the scheme is $$O(10^{-8})$$ and the physical structure of the obtained solutions are shown by graphic illustration in order to give a better understanding on the dynamics of both optical dromions and domain walls.

Chaos in Bohmian Quantum Mechanics: A Short Review

Abstract: This is a short review of the theory of chaos in Bohmian quantum mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally, we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i. e., they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tend to Born’s rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born’s rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born’s rule unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian mechanics, namely, whether there is a dynamical approximation of Born’s rule by an arbitrary initial distribution of Bohmian particles.

Dynamics of Rubber Chaplygin Sphere under Periodic Control

Abstract: This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant submanifolds corresponding to plane-parallel motion of the sphere with rotation about the minor, middle and major axes of inertia. The above-mentioned motions are quasi-periodic, and for the numerical estimate of their stability charts of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and amplitude of the moments of inertia. It is shown that rotations about the minor and major axes of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In this case, the so-called “Arnol’d tongues” arise in the stability chart. Stabilization of the middle unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It is shown that the nonconservativeness of the system manifests itself in the occurrence of limit cycles, attracting tori and strange attractors in phase space. Numerical calculations show that strange attractors may arise through a cascade of period-doubling bifurcations or after a finite number of torus-doubling bifurcations.

Dynamics of the Tippe Top on a Vibrating Base

Abstract: This paper studies the conditions under which the tippe top inverts in the presence of vibrations of the base along the vertical. A vibrational potential is constructed by averaging and it is shown that, when this potential is added to the system, the Jellett integral is preserved. This makes it possible to apply the modified Routh method and to find the effective potential to whose critical points permanent rotations or regular precessions of the tippe top correspond. Tippe top inversion is possible for a sufficiently large initial angular velocity under the condition that spinning with the lowest position of the center of gravity is unstable, spinning with the highest position of the center of gravity is stable, and that there are no precessions. Cases are found in which there is no inversion in the absence of vibrations, but it can be brought about by a suitable choice of the mean value of the squared velocity of the base. In particular, this type includes a ball with a spherical cavity filled with a denser substance.

Persistence of Hyperbolic-type Degenerate Lower-dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems

Abstract: It is known that under Kolmogorov’s nondegeneracy condition, the nondegenerate hyperbolic invariant torus with Diophantine frequencies will persist under small perturbations, meaning that the perturbed system still has an invariant torus with prescribed frequencies. However, the degenerate torus is sensitive to perturbations. In this paper, we prove the persistence of two classes of hyperbolic-type degenerate lower-dimensional invariant tori, one of them corrects an earlier work [34] by the second author. The proof is based on a modified KAM iteration and analysis of stability of degenerate critical points of analytic functions.

Dynamics and Bifurcations on the Normally Hyperbolic InvariantManifold of a Periodically Driven System with Rank-1 Saddle

Abstract: In chemical reactions, trajectories typically turn from reactants to products when crossing a dividing surface close to the normally hyperbolic invariant manifold (NHIM) given by the intersection of the stable and unstable manifolds of a rank-1 saddle. Trajectories started exactly on the NHIM in principle never leave this manifold when propagated forward or backward in time. This still holds for driven systems when the NHIM itself becomes time-dependent. We investigate the dynamics on the NHIM for a periodically driven model system with two degrees of freedom by numerically stabilizing the motion. Using Poincare surfaces of section, we demonstrate the occurrence of structural changes of the dynamics, viz., bifurcations of periodic transition state (TS) trajectories when changing the amplitude and frequency of the external driving. In particular, periodic TS trajectories with the same period as the external driving but significantly different parameters — such as mean energy — compared to the ordinary TS trajectory can be created in a saddle-node bifurcation.

Stability and Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base

Abstract: This paper addresses the problem of a spherical robot having an axisymmetric pendulum drive and rolling without slipping on a vibrating plane. It is shown that this system admits partial solutions (steady rotations) for which the pendulum rotates about its vertical symmetry axis. Special attention is given to problems of stability and stabilization of these solutions. An analysis of the constraint reaction is performed, and parameter regions are identified in which a stabilization of the spherical robot is possible without it losing contact with the plane. It is shown that the partial solutions can be stabilized by varying the angular velocity of rotation of the pendulum about its symmetry axis, and that the rotation of the pendulum is a necessary condition for stabilization without the robot losing contact with the plane.

The Tippedisk: a Tippetop Without Rotational Symmetry

Abstract: The aim of this paper is to introduce the tippedisk to the theoretical mechanics community as a new mechanical-mathematical archetype for friction induced instability phenomena. We discuss the modeling and simulation of the tippedisk, which is an inhomogeneous disk showing an inversion phenomenon similar but more complicated than the tippetop. In particular, several models with different levels of abstraction, parameterizations and force laws are introduced. Moreover, the numerical simulations are compared qualitatively with recordings from a high-speed camera. Unlike the tippetop, the tippedisk has no rotational symmetry, which greatly complicates the three-dimensional nonlinear kinematics. The governing differential equations, which are presented here in full detail, describe all relevant physical effects and serve as a starting point for further research.

Kovalevskaya Exponents, Weak Painlevé Property and Integrability for Quasi-homogeneous Differential Systems

Abstract: We present some necessary conditions for quasi-homogeneous differential systems to be completely integrable via Kovalevskaya exponents. Then, as an application, we give a new link between the weak-Painleve property and the algebraical integrability for polynomial differential systems. Additionally, we also formulate stronger theorems in terms of Kovalevskaya exponents for homogeneous Newton systems, a special class of quasi-homogeneous systems, which gives its necessary conditions for B-integrability and complete integrability. A consequence is that the nonrational Kovalevskaya exponents imply the nonexistence of Darboux first integrals for two-dimensional natural homogeneous polynomial Hamiltonian systems, which relates the singularity structure to the Darboux theory of integrability.

On Dynamics of Jellet’s Egg. Asymptotic Solutions Revisited

Abstract: We study here the asymptotic condition $$\dot E = - \mu {g_n}b_A^2 = 0$$ for an eccentric rolling and sliding ellipsoid with axes of principal moments of inertia directed along geometric axes of the ellipsoid, a rigid body which we call here Jellett’s egg (JE). It is shown by using dynamic equations expressed in terms of Euler angles that the asymptotic condition is satisfied by stationary solutions. There are 4 types of stationary solutions: tumbling, spinning, inclined rolling and rotating on the side solutions. In the generic situation of tumbling solutions concise explicit formulas for stationary angular velocities $${\dot \varphi _{{\rm{JE}}}}(\cos \,\theta),\,{\omega _{3{\rm{JE}}}}(\cos \,\theta)$$ as functions of JE parameters $$\widetilde\alpha ,\,\alpha ,\,\gamma $$ are given. We distinguish the case $$1 - \widetilde\alpha < {\alpha ^2} < 1 + \widetilde{\alpha ,}\,1 - \widetilde{\alpha ,} < {\alpha ^2}\gamma < 1 + \widetilde\alpha $$ when velocities $${\varphi _{{\rm{JE}}}},{\omega _3}_{{\rm{JE}}}$$ are defined for the whole range of inclination angles θ ∈ (0, π). Numerical simulations illustrate how, for a JE launched almost vertically with $$\theta \left(0 \right) = {1 \over {100}},\,{1 \over {10}}$$, the inversion of the JE depends on relations between parameters.

The Role of Depth and Flatness of a Potential Energy Surface in Chemical Reaction Dynamics

Abstract: In this study, we analyze how changes in the geometry of a potential energy surface in terms of depth and flatness can affect the reaction dynamics. We formulate depth and flatness in the context of one- and two-degree-of-freedom (DOF) Hamiltonian normal form for the saddle-node bifurcation and quantify their influence on chemical reaction dynamics [1, 2]. In a recent work, Garcia-Garrido et al. [2] illustrated how changing the well-depth of a potential energy surface (PES) can lead to a saddle-node bifurcation. They have shown how the geometry of cylindrical manifolds associated with the rank-1 saddle changes en route to the saddle-node bifurcation. Using the formulation presented here, we show how changes in the parameters of the potential energy control the depth and flatness and show their role in the quantitative measures of a chemical reaction. We quantify this role of the depth and flatness by calculating the ratio of the bottleneck width and well width, reaction probability (also known as transition fraction or population fraction), gap time (or first passage time) distribution, and directional flux through the dividing surface (DS) for small to high values of total energy. The results obtained for these quantitative measures are in agreement with the qualitative understanding of the reaction dynamics.

Revisiting the Human and Nature Dynamics Model

Abstract: We present a simple model for describing the dynamics of the interaction between a homogeneous population or society, and the natural resources and reserves that the society needs for its survival. The model is formulated in terms of ordinary differential equations, which are subsequently discretised, the discrete system providing a natural integrator for the continuous one. An ultradiscrete, generalised cellular automaton-like, model is also derived. The dynamics of our simple, three-component, model are particularly rich exhibiting either a route to a steady state or an oscillating, limit cycle-type regime or to a collapse. While these dynamical behaviours depend strongly on the choice of the details of the model, the important conclusion is that a collapse or near collapse, leading to the disappearance of the population or to a complete transfiguration of its societal model, is indeed possible.

Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems

Abstract: We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with a strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all Casimirs linear in momenta on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are either constant or periodic. Some related results for other Carnot groups are presented.

The Method of Averaging for the Kapitza - Whitney Pendulum

Abstract: A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i. e., the pendulum never falls, provided the period of vibration and the period of horizontal force are commensurable. We also present a sufficient condition for the existence of at least two different periodic solutions without falling. We show numerically that there exist stable periodic solutions without falling.

Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada — Kotera and Kupershmidt Equations

Abstract: Self-similar reductions of the Sawada-Kotera and Kupershmidt equations are studied. Results of Painleve’s test for these equations are given. Lax pairs for solving the Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions of the Sawada-Kotera and Kupershmidt equations expressed via the first Painleve equation are presented. Exact solutions of the Sawada-Kotera and Kupershmidt equations by means of general solution for the first member of K2 hierarchy are given. Special polynomials for expressions of rational solutions for the equations considered are introduced. The differential-difference equations for finding special polynomials corresponding to the Sawada-Kotera and Kupershmidt equations are found. Nonlinear differential equations of sixth order for special polynomials associated with the Sawada-Kotera and Kupershmidt equations are obtained. Lax pairs for nonlinear differential equations with special polynomials are presented. Rational solutions of the self-similar reductions for the Sawada-Kotera and Kupershmidt equations are given.

Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass

Abstract: The motion of a spherical robot with periodically changing moments of inertia, internal rotors and a displaced center of mass is considered. It is shown that, under some restrictions on the displacement of the center of mass, the system of interest features chaotic dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the case of fixed rotors and periodically changing moments of inertia.

Rational Solutions of Equations Associated with the Second Painlevé Equation

Abstract: Nonlinear differential equations associated with the second Painleve equation are considered. Transformations for solutions of the singular manifold equation are presented. It is shown that rational solutions of the singular manifold equation are determined by means of the Yablonskii-Vorob’ev polynomials. It is demonstrated that rational solutions for some differential equations are also expressed via the Yablonskii-Vorob’ev polynomials.

On the Stability of Potential Systems under the Action of Non-conservative Positional Forces

Abstract: The stability of linear mechanical systems with finite numbers of degrees of freedom subjected to potential and non-conservative positional forces is considered. The positive semi-definiteness of the potential energy is assumed. Three new stability criteria which are in a simple way related to the properties of the system matrices are derived. These criteria improve previously obtained results of the same type. Several simple examples are given to illustrate the correctness and applicability of the obtained results.

Rheonomic Systems with Nonlinear Nonholonomic Constraints: The Voronec Equations

Abstract: One of the earliest formulations of dynamics of nonholonomic systems traces back to 1895 and is due to Chaplygin, who developed his analysis under the assumption that a certain number of the generalized coordinates do not occur either in the kinematic constraints or in the Lagrange function. A few years later Voronec derived equations of motion for nonholonomic systems removing the restrictions demanded by the Chaplygin systems. Although the methods encountered in the following years favor the use of the quasi-coordinates, we will pursue the Voronec method, which deals with the generalized coordinates directly. The aim is to establish a procedure for extending the equations of motion to nonlinear nonholonomic systems, even in the rheonomic case.

Nonequilibrium Molecular Dynamics, Fractal Phase-Space Distributions, the Cantor Set,and Puzzles Involving Information Dimensions for Two Compressible Baker Maps

Abstract: Deterministic and time-reversible nonequilibrium molecular dynamics simulations typically generate “fractal” (fractional-dimensional) phase-space distributions. Because these distributions and their time-reversed twins have zero phase volume, stable attractors “forward in time” and unstable (unobservable) repellors when reversed, these simulations are consistent with the second law of thermodynamics. These same reversibility and stability properties can also be found in compressible baker maps, or in their equivalent random walks, motivating their careful study. We illustrate these ideas with three examples: a Cantor set map and two linear compressible baker maps, N2 $$(q,p)$$ and N3 $$(q,p)$$ . The two baker maps’ information dimensions estimated from sequential mappings agree, while those from pointwise iteration do not, with the estimates dependent upon details of the approach to the maps’ nonequilibrium steady states.

Nondegenerate Hamiltonian Hopf Bifurcations in $$\omega:3:6$$ Resonance $$(\omega=1$$ or $$2)$$

Abstract: This paper deals with the analysis of Hamiltonian Hopf bifurcations in three-degree-of-freedom systems, for which the frequencies of the linearization of the corresponding Hamiltonians are in $$\omega:3:6$$ resonance ( $$\omega=1$$ or $$2$$ ). We obtain the truncated second-order normal form that is not integrable and expressed in terms of the invariants of the reduced phase space. The truncated first-order normal form gives rise to an integrable system that is analyzed using a reduction to a one-degree-of-freedom system. In this paper, some detuning parameters are considered and nondegenerate Hamiltonian Hopf bifurcations are found. To study Hamiltonian Hopf bifurcations, we transform the reduced Hamiltonian into standard form.

Confinement Strategies in a Simple SIR Model

Abstract: We propose a simple deterministic, differential equation-based, SIR model in order to investigate the impact of various confinement strategies on a most virulent epidemic. Our approach is motivated by the current COVID-19 pandemic. The main hypothesis is the existence of two populations of susceptible persons, one which obeys confinement and for which the infection rate does not exceed 1, and a population which, being non confined for various imperatives, can be substantially more infective. The model, initially formulated as a differential system, is discretised following a specific procedure, the discrete system serving as an integrator for the differential one. Our model is calibrated so as to correspond to what is observed in the COVID-19 epidemic, for the period from February 19 to April 16. Several conclusions can be reached, despite the very simple structure of our model. First, it is not possible to pinpoint the genesis of the epidemic by just analysing data from when the epidemic is in full swing. It may well turn out that the epidemic has reached a sizeable part of the world months before it became noticeable. Concerning the confinement scenarios, a universal feature of all our simulations is that relaxing the lockdown constraints leads to a rekindling of the epidemic. Thus, we sought the conditions for the second epidemic peak to be lower than the first one. This is possible in all the scenarios considered (abrupt or gradualexit, the latter having linear and stepwise profiles), but typically a gradual exit can start earlier than an abrupt one. However, by the time the gradual exit is complete, the overall confinement times are not too different. From our results, the most promising strategy is that of a stepwise exit. Its implementation could be quite feasible, with the major part of the population (perhaps, minus the fragile groups) exiting simultaneously, but obeying rigorous distancing constraints.

Quantifying the Transition from Spiral Waves to Spiral Wave Chimeras in a Lattice of Self-sustained Oscillators

Abstract: The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the network under consideration are the van der Pol oscillator or the FitzHugh – Nagumo neuron. Both of the models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity, which enables us to evaluate the sensitivity of each oscillator at a finite time. Spiral waves are observed in both lattices when the interaction between elements has a local character. The dynamics of all the elements is regular. There are no pronounced high-sensitive regions. We have discovered that, when the coupling becomes nonlocal, the features of the system change significantly. The oscillation regime of the spiral wave center element switches to a chaotic one. Besides, a region with high sensitivity occurs around the wave center oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this high-sensitive area. A sharp increase in the values of the maximal Lyapunov exponent in the positive region leads to the formation of the incoherence cluster. Furthermore, we find that the system can even switch to a hyperchaotic regime when several Lyapunov exponents become positive.

On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem

Abstract: The well-known problem of the nonlinear stability of L4 and L5 in the circular spatial restricted three-body problem is revisited. Some new results in the light of the concept of Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates for three specific values of the mass ratio that remained uncovered. Moreover, in many cases we improve the estimates found in the literature.

Two Nonholonomic Chaotic Systems. Part I. On the Suslov Problem

Abstract: A generalization of the Suslov problem with changing parameters is considered. The physical interpretation is a Chaplygin sleigh moving on a sphere. The problem is reduced to the study of a two-dimensional system describing the evolution of the angular velocity of a body. The system without viscous friction and the system with viscous friction are considered. Poincare maps are constructed, attractors and noncompact attracting trajectories are found. The presence of noncompact trajectories in the Poincare map suggests that acceleration is possible in this nonholonomic system. In the case of a system with viscous friction, a chart of dynamical regimes and a bifurcation tree are constructed to analyze the transition to chaos. The classical scenario of transition to chaos through a cascade of period doubling is shown, which may indicate attractors of Feigenbaum type.

On Topological Classification of Gradient-like Flows on an $$n$$ -sphere in the Sense of Topological Conjugacy

Abstract: In this paper, we study gradient-like flows without heteroclinic intersections on an $$n$$ -sphere up to topological conjugacy. We prove that such a flow is completely defined by a bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to the topological equivalence also. This result implies that for these flows with the same (up to a change of coordinates) partitions into trajectories, the partitions for elements, composing isotopies connecting time-one shifts of these flows with the identity map, also coincide. This phenomenon strongly contrasts with the situation for flows with periodic orbits and connections, where one class of equivalence contains continuum classes of conjugacy. In addition, we realize every connected bicolor tree by a gradient-like flow without heteroclinic intersections on the $$n$$ -sphere. In addition, we present a linear-time algorithm on the number of vertices for distinguishing these trees.