Search

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves. [ABSTRACT FROM AUTHOR]

In this work, we present some criteria about the existence and nonexistence of both Puiseux inverse integrating factors V$V$ and Puiseux first integrals H$H$ for planar analytic vector fields having a monodromic singularity. These functions are a wide generalization of their formal R[[x,y]]$\mathbb {R}[[x,y]]$ or algebraic counterpart in Cartesian coordinates (x,y)$(x,y)$. We prove that none of the functions H$H$ and V$V$ can be used to characterize degenerate centers although the existence of H$H$ is a sufficient center condition. [ABSTRACT FROM AUTHOR]

In the last decade the flux-charge analysis method (FCAM) has been successfully used to show that continuous-time (CT) memristor circuits possess for structural reasons first integrals (invariants of motion) and their state space can be foliated in invariant manifolds. Consequently, they display an initial condition dependent dynamics, extreme multistability (coexistence of infinitely many attractors) and bifurcations without parameters. Recently, a new discretization scheme has been introduced for CT memristor circuits, guaranteeing that the first integrals are preserved exactly in the discretization. On this basis, FCAM has been extended to discrete-time (DT) memristor circuits showing that they also are characterized by invariant manifolds and they display extreme multistability and bifurcations without parameters. This manuscript considers the maps obtained via DT-FCAM for a circuit with a flux-controlled memristor and a capacitor and it provides a thorough and rigorous investigation of the presence of chaotic dynamics. In particular, parameter ranges are obtained where the maps have snap-back repellers at some fixed points, thus implying that they display chaos in the Marotto and also in the Li–Yorke sense. Bifurcation diagrams are provided where it is possible to analytically identify relevant points in correspondence with the appearance of snap-back repellers and the onset of chaos. The dependence of the bifurcation diagrams and snap-back repellers upon the circuit initial conditions and the related manifold is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere S 2 = { (x , y , z) ∈ R 3 | x 2 + y 2 + z 2 = 1 } . We start by classifying all degree three polynomial vector fields on S 2 and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on S 2 and also study the maximum number of various types of invariant great circles for homogeneous cubic vector fields on S 2 . We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on S 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study the maximum number of limit cycles where planar discontinuous piecewise differential systems can exist formed by three regions separated by a nonregular line. We show that such discontinuous piecewise linear systems can have at most five limit cycles, two of which are of the four intersection points type, the third one is of the two intersection points type and the other two are of three intersection points type. In a setting, we have solved the extended 16th Hilbert problem for these discontinuous piecewise linear differential systems. [ABSTRACT FROM AUTHOR]

One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations [R. Moussu, Une démonstration géométrique d'un théorème de Lyapunov-Poincaré, Astérisque 98–99 (1982), pp. 216–223]. In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with 'many' periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework. [ABSTRACT FROM AUTHOR]

We study the nonholonomic motion of a point particle on the Heisenberg group around the fixed “sun” placed at the origin whose potential is given by the fundamental solution of the sub-Laplacian. In contrast with several recent papers that approach this problem as a variational one (hence a control problem) we study the equations of dynamical motion which are non-variational in nonholonomic mechanics. We find three independent first integrals of the system and show that its bounded trajectories are wound up around certain surfaces of the fourth order. We also describe some particular cases of trajectories. [ABSTRACT FROM AUTHOR]

Approximate streamsurfaces of a three-dimensional velocity field have recently been constructed as isosurfaces of the closest first integral of the velocity field. Such approximate streamsurfaces enable effective and efficient visualization of vortical regions in three-dimensional flows. Here we propose a variational construction of these approximate streamsurfaces to remove the limitation of Fourier series representation of the first integral in earlier work. Specifically, we use finite-element methods to solve a partial differential equation that describes the best approximate first integral for a given velocity field. We use several examples to demonstrate the power of our approach for three-dimensional flows in domains with arbitrary geometries and boundary conditions. These include generalized axisymmetric flows in the domains of a sphere (spherical vortex), a cylinder (cylindrical vortex) and a hollow cylinder (Taylor–Couette flow) as benchmark studies for various computational domains, non-integrable periodic flows (ABC and Euler flows) and Rayleigh–Bénard convection flows. We also illustrate the use of the variational construction in extracting momentum barriers in Rayleigh–Bénard convection. [ABSTRACT FROM AUTHOR]

In the last few years the literature has witnessed a remarkable surge of interest for maps implemented by discrete-time (DT) memristor circuits. One main reason is that from numerical simulations it appears that even for simple memristor circuits the maps can easily display complex dynamics, including chaos and hyperchaos, which are of relevant interest for engineering applications. Goal of this manuscript is to investigate on the reasons underlying this type of complex behavior. To this end, the manuscript considers the map implemented by the simplest memristor circuit given by a capacitor and an ideal flux-controlled memristor or an inductor and an ideal charge-controlled memristor. In particular, the manuscript uses the DT flux-charge analysis method (FCAM) introduced in a recent paper to ensure that the first integrals and foliation in invariant manifolds of continuous-time (CT) memristor circuits are preserved exactly in the discretization for any step size. DT-FCAM yields a two-dimensional map in the voltage-current domain (VCD) and a manifold-dependent one-dimensional map in the flux-charge domain (FCD), i.e., a one-dimensional map on each invariant manifold. One main result is that, for suitable choices of the circuit parameters and memristor nonlinearities, both DT circuits can exactly embed two classic chaotic maps, i.e., the logistic map and the tent map. Moreover, due to the property of extreme multistability, the DT circuits can simultaneously embed in the manifolds all the dynamics displayed by varying one parameter in the logistic and tent map. The paper then considers a DT memristor Murali-Lakshmanan-Chua circuit and its dual. Via DT-FCAM these circuits implement a three-dimensional map in the VCD and a two-dimensional map on each invariant manifold in the FCD. It is shown that both circuits can simultaneously embed in the manifolds all the dynamics displayed by two other classic chaotic maps, i.e., the Hénon map and the Lozi map, when varying one parameter in such maps. In essence, these results provide an explanation of why it is not surprising to observe complex dynamics even in simple DT memristor circuits.

We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we have solved the extension of the 16th Hilbert problem to this class of differential systems.

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua's circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds. [ABSTRACT FROM AUTHOR]

We consider the symmetries of partial differential equations based on the use of differential-geometric and algebraic methods of the theory of dynamical control systems. [ABSTRACT FROM AUTHOR]

This article explores the generalized Gerdjikov–Ivanov equation describing the propagation of pulses in optical fiber. The equation studied has a variety of applications, for instance, in photonic crystal fibers. In contrast to the classical Gerdjikov–Ivanov equation, the solution of the Cauchy problem for the studied equation cannot be found by the inverse scattering problem method. In this regard, analytical solutions for the generalized Gerdjikov–Ivanov equation are found using traveling-wave variables. Phase portraits of an ordinary differential equation corresponding to the partial differential equation under consideration are constructed. Three conservation laws for the generalized equation corresponding to power conservation, moment and energy are found by the method of direct transformations. Conservative densities corresponding to optical solitons of the generalized Gerdjikov–Ivanov equation are provided. The conservative quantities obtained have not been presented before in the literature, to the best of our knowledge. [ABSTRACT FROM AUTHOR]

In many real-world systems, the entry rate of particles into a lane is affected by the occupancy of nearby pools. For instance, in biological networks, the concentration of molecules on the side of a membrane affects the entry of particles through the membrane. To understand the behaviour of such networks, we develop a network model of ribosome flow models (RFMs) having multiple pools where each RFM captures the dynamics of particle flow in a lane and competes for the finite resources present at the nearby pool. We study a ribosome flow model network with two pools (RFMNTP) and show that the network always admits a steady state. We then analyse the behaviour of the RFMNTP with respect to modifying the transition rate through a theoretical framework. Simulations of the RFMNTP demonstrate a counterintuitive result. For example, increasing any of the transition rates in the presence of a slow site in an RFM can increase the output rate of some RFMs and decrease the output rate of the other RFMs simultaneously. This suggests that the role of local sharing of particles incorporated is non-trivial. Finally, we illustrate how these results can provide insights into studying a network with multiple pools.

The current research aims to analyze the Ivancevic option pricing model and investigate its soliton solutions, which offer a comprehensive understanding of various related phenomena. The equation's Cauchy problem cannot be resolved through the inverse scattering transform, necessitating an analytical approach for accurate traveling wave solutions. To achieve this, the study employs the generalized logistic equation method and Nucci's reduction approaches to derive solitary wave solutions. Moreover, Nucci's reduction procedure is applied to formulate the first integral of a differential equation, ensuring conservation and exact solutions. In order to gauge the Ivancevic option pricing equation's sensitivity, a sensitivity analysis is conducted. At the end, the chaotic behavior of the considered model with the perturbed term is examined by using Lyapunov exponent approach.

The integrability of fourth‐order ordinary differential equations admitting gl(2,ℝ) as Lie symmetry algebra is addressed in this work. The classical Lie's method of reduction cannot be applied to solve by quadrature this kind of equations because gl(2,ℝ) is nonsolvable. In order to avoid such difficulty, a solvable structure involving the vector field identified with the equation is constructed by using the symmetry generators of gl(2,ℝ). This permits to compute a first integral of the equation by quadrature. In the aftermath, it is shown that the general solution of any GL(2,ℝ)‐invariant fourth‐order ordinary differential equation can be obtained in parametric form, involving linearly independent solutions to a related one‐parameter family of linear second‐order equations. Particular examples are also shown with the end of illustrating the presented approach. [ABSTRACT FROM AUTHOR]

We study one-parameter analytic integrable deformations of the germ of 2 × (n − 2)-type complex saddle singularity given by d(xy) = 0 at the origin 0 ∈ ℂ 2 × ℂ n − 2 , n ≥ 2 . Such a deformation writes ω t = d (x y) + ∑ j = 1 ∞ t j ω j where t ∈ (ℂ , 0) is the parameter of the deformation and the coefficients ωj are holomorphic one-forms in some neighborhood of the origin 0 ∈ ℂ n . We consider a natural condition on the singular set of the deformation with respect to the fibration d(xy) = 0. Under this condition, the existence of a holomorphic first integral for each element ωt of the deformation is equivalent to the vanishing of certain line integrals ∮ γ c ω t = 0 , ∀ γ c , ∀ t calculated on cycles γc contained in the fibers (x y = c) , 0 ≠ c ∈ (ℂ , 0) . This result is quite sharp regarding the conditions of the singular set and on the vanishing of the integrals in cycles. It is also not valid for ramified saddles, i.e., for deformations of saddles of the form xnym = c where n + m > 2. As an application of our techniques we obtain a criteria for the existence of first integrals for integrable codimension one deformations of quadratic real analytic center-cylinder type singularities in terms of the vanishing of some easy to compute line integrals. [ABSTRACT FROM AUTHOR]

We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we have solved the extension of the 16th Hilbert problem to this class of differential systems. [ABSTRACT FROM AUTHOR]

The problem of indirect single-position coordinate determination based on the smoothed measurements of bearing and the radial velocity of an object is solved considering motion invariants and singular measurement errors. Such errors are represented as an appropriate linear combination with unknown spectral coefficients in a given finite-dimensional functional space. Possible application of the developed method to different models of motion and observation is considered. Analytical relations are derived for estimating accuracy characteristics and methodological errors. A comparative evaluation of computational cost is presented. [ABSTRACT FROM AUTHOR]

Although Lyapunov exponents have been widely used to characterize the dynamics of nonlinear systems, few methods are available so far to obtain a priori bounds on their magnitudes. Recently, sufficient conditions to rule out the existence of attractors with positive Lyapunov exponents have been derived via a Lyapunov approach based on the second additive compound matrices of the system Jacobian. This paper first provides some insights into this approach by showing how the several available techniques for computing Lyapunov functions can be fruitfully applied to Lorenz and Thomas systems to derive explicit conditions on their system parameters, which ensure that there are no attractors with positive Lyapunov exponents. Then, the approach is extended to the case of nonlinear systems with a first integral of motion and its application to the memristor Chua's circuit is discussed. [ABSTRACT FROM AUTHOR]

We exhibit a class of non-linear planar polynomial differential systems. We show that this class has a first integral and we give the explicit expression of them and we discuss the existence of two limit cycles surrounding an unstable node according to the parameters of these systems. [ABSTRACT FROM AUTHOR]

The present work provides an analytical treatment of a modified version of the Lindemann mechanism with three reaction rate constants. We firstly derive the exact analytical expressions among the three concentrations, then based on which the specific analytical forms of the invariant lines and feasible regions are also obtained, although the existence of the invariant lines has been proved in a recent work. An efficient semi-numerical and semi-analytical solution is also constructed in the framework of Piecewise differential transform method (PDTM). What is more, the reaction dynamical properties including the limiting behaviors are further discussed. [ABSTRACT FROM AUTHOR]

We study local integrability and linearizability of polynomial and analytic systems of ODEs. It is proven that in the case of non-degenerate singularity if an analytic system is completely integrable and one equation is linearizable, then the system is linearizable in a neighborhood of the singularity. Some integrable and linearizable systems in a family of three-dimensional quadratic autonomous systems of ODEs depending on ten parameters are found. [ABSTRACT FROM AUTHOR]

Because of their applications to many physical phenomena in recent decades, interest in the study of piecewise discontinuous differential systems has increased greatly. Limit cycles play an important role in the study of any planar differential system, but determining the maximum number of limit cycles that a class of planar differential systems can have is one of the main problems in the qualitative theory of planar differential systems. Thus, in general, providing a precise upper bound for the number of crossing limit cycles that a given class of piecewise linear differential systems can have is a very difficult problem. In this paper, we provide the exact upper bound of limit cycles for linear piecewise differential systems, formed by linear Hamiltonian systems without equilibrium and separated by a two concentric circles. Furthermore, we prove that our result is reached by giving some systems having exactly one, two or three limit cycles. [ABSTRACT FROM AUTHOR]

This work presents a generalized hyper-elastic rod wave (gHRW) equation from the Lie symmetry method’s standpoint. The equation illustrates dispersive waves generating in hyper-elastic rods. Using multiplier approach we find conserved vectors of the underlying equation. We subsequently obtain first integrals of the conserved vectors under the time–space group invariant u(t,x)=H(x−νt). Finally, by analysing various attainable instances of the arbitrary coefficient function g(u), we perform symmetry reductions of gHRW equation to lower order ordinary differential equations and in some instances obtain analytic solutions for special values of arbitrary constants.

The objective of this study is to investigate a few solutions to the nonlinear Schrödinger problem with parabolic law. The first integral and exact solutions for the reduced ODE of the model under consideration are extracted using Nucci’s reduction approach. Finally, using the efficient and effective solutions technique, we display density plots and 2D, 3D plots for the suggested governing model.

This study presents a novel reduction method to obtain exact solutions for the behavior of a porous fin under a uniform magnetic field, considering the effects of convection, radiation, and internal heat generation. The study applies the reduction method to simplify the governing equations and reduce the complexity of the mathematical model, making it possible to obtain closed-form solutions. The results of the analysis provide a comprehensive understanding of the behavior of the porous fin, including the temperature distribution and heat transfer characteristics, and the impact of the magnetic field on its performance. The findings of this research could be useful for the analysis and description of various heat transfer and energy systems that involve porous fins in magnetic fields.

This paper shows an oscillator with a spring made of material where the stress is a function not only of strain but also strain rate. The corresponding restitution force is of strong nonlinear monomial type and is the product of displacement and velocity of any order. The mathematical model of the oscillator is a homogenous strong nonlinear second-order differential equation with an integer- or non-integer-order mixed term. In the paper, an analytical procedure for solving this new type of strong nonlinear equation is developed. The approximate solution is assumed as the perturbed version of the exact solution in the form of a sine Ateb function. As a result, it is obtained that the amplitude, period, and frequency of vibration depend not only on the coefficient and order of nonlinearity, but also on the initial velocity. The procedure is tested on two examples: oscillator perturbed with small linear damping and small linear displacement functions. The analytically obtained results are compared with the exact numerical ones and show good agreement. It is concluded that the mathematical model and also the procedure developed in the paper would be convenient for prediction of motion for this type of oscillator without necessary experimental testing. [ABSTRACT FROM AUTHOR]

A simple SIS-type mathematical model of infection expansion is presented and analysed with focus on the case SARS-Cov-2. It takes into account two processes, namely, infection and recovery/decease characterised by two parameters in total: contact rate and recovery/decease rate. Its solution has a form of a quasi-logistic function for which we have introduced an infection index that, should it become negative, can also be considered as a recovery/decease index with decrease of infected down to zero. Based on the data from open sources for the SARS-Cov-2 pandemic, seasonal influenza epidemics and a pandemic in the fauna world, a threshold value of the infection index has been shown to exist above which an infection expansion pretends to be considered as pandemic. Lean (two-parameter) SIR models affined with the warning SIS model have been built. Their general solutions have been obtained, analysed and shown to be a priori structurally adjusted to the infectives' peak in epidemiological data. [ABSTRACT FROM AUTHOR]

The Painlevé-Ince differential equation y ′ ′ + 3 y y ′ + y 3 = 0 has been studied from many points of view. Here we complete its study providing its phase portrait in the Poincaré disc. [ABSTRACT FROM AUTHOR]

The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the (1 + 1) − dimensional Schrödinger dynamical equation, which is the subject of this study. New solitary wave profiles are discovered by using Nucci's scheme and a new extended direct algebraic method. The new extended direct algebraic approach provides an easy and general mechanism for covering 37 solitonic wave solutions, which roughly corresponds to all soliton families, and Nucci's direct reduction method is used to develop the first integral and the exact solution of partial differential equations. Thus, there are several new solitonic wave patterns that are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, a mixed trigonometric solution, a trigonometric solution, a shock solution, a mixed shock singular solution, a mixed singular solution, a complex solitary shock solution, a singular solution, and shock wave solutions. The first integral of the considered model and the exact solution are obtained by utilizing Nucci's scheme. We present 2-D, 3-D, and contour graphics of the results obtained to illustrate the pulse propagation characteristics while taking suitable values for the parameters involved, and we observed the influence of parameters on solitary waves. It is noticed that the wave number α and the soliton speed μ are responsible for controlling the amplitude and periodicity of the propagating wave solution. [ABSTRACT FROM AUTHOR]

In 2010, Muthuswamy and Chua presented an autonomous chaotic circuit using only three elements in series: an inductor, a capacitor and a memristor. This circuit is known as the simplest chaotic circuit and it is determined by a three-dimensional differential system, which depends on the real parameters C, L, α and β. Although the Muthuswamy-Chua system is simpler in formulation than other chaotic systems, its dynamics has proven to be complicated. Here we analytically prove the existence of periodic orbits in this system for suitable choice of the parameter values α and β leading to interesting phenomena as multistability and formation of chaotic attractors. In order to do that, we consider the existence of first integrals, invariant algebraic surfaces and a result from averaging theory. In addition, we relate the obtained results to the memristance and to the physical characteristics of the memristor. [ABSTRACT FROM AUTHOR]

We study germs of holomorphic distributions with "separated variables". In codimension one, a well know example of this kind of distribution is given by d z = (y 1 d x 1 - x 1 d y 1) + ⋯ + (y m d x m - x m d y m) , which defines the canonical contact structure on C P 2 m + 1 . Another example is the Darboux distribution d z = x 1 d y 1 + ⋯ + x m d y m , which gives the normal local form of any contact structure. Given a germ D of holomorphic distribution with separated variables in (C n , 0) , we show that there exists , for some κ ∈ Z ≥ 0 related to the Taylor coefficients of D , a holomorphic submersion H D : (C n , 0) → (C κ , 0) such that D is completely non-integrable on each level of H D . Furthermore, we show that there exists a holomorphic vector field Z tangent to D , such that each level of H D contains a leaf of Z that is somewhere dense in the level. In particular, the field of meromorphic first integrals of Z and that of D are the same. Between several other results, we show that the canonical contact structure on C P 2 m + 1 supports a Legendrian holomorphic foliation whose generic leaves are dense in C P 2 m + 1 . So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense. [ABSTRACT FROM AUTHOR]

We consider the continuous version of the nonlinear Hirota oscillator equation and by using the Jacobi last multiplier (JLM) show that two types of Lagrangians (or Hamiltonians) can be derived corresponding to different choices of the JLM. The existence of a mirror partner (or sister) equation is shown. Both equations belong to the Liénard-II class and the Hirota oscillator equations are shown to emerge from the requirement of isochronicity. [ABSTRACT FROM AUTHOR]

The study of discontinuous piecewise differential systems has attracted increasing attention in recent decades as a result of their applications to a variety of physical phenomena. The role of limit cycles in the study of any planar differential system is well known, but finding the maximum number of limit cycles that a class of planar differential systems can have is one of the most difficult tasks in the qualitative theory of planar differential systems. Thus, in this work, we solve the extinction of Hilbert problem for all classes of discontinuous piecewise differential systems with Hamiltonian nilpotent saddles, separated by the straight line x = 0 . Firstly, we study the discontinuous piecewise differential system formed by linear center and one of the six Hamiltonian nilpotent saddles, we provide that these systems can have at most one limit cycle. Secondly, we consider all possible discontinuous piecewise differential systems formed by Hamiltonian nilpotent saddles in each piece, where we show that these 21 classes of piecewise differential systems can have at most four limit cycles. We have strengthened our result by giving examples for each case. [ABSTRACT FROM AUTHOR]

The first three members of the nonlinear ordinary differential equations for the derivative nonlinear Schrödinger hierarchy are considered. Reduction to nonlinear ordinary differential equations is utilized to obtain traveling wave solutions. Lax pairs for these nonlinear ordinary differential equations are presented. The first integrals of nonlinear ordinary differential equations are obtained by taking Lax pairs into account. Exact solutions in the form of solitary and periodic waves are found by means of the generalized method of auxiliary equations. • The traveling wave reduction of the Kaup-Newell hierarchy is studied. • The Lax pair associated with this hierarchy is found. • The first integrals of nonlinear ordinary differential equations are obtained. • Exact solutions in the form of solitary and periodic waves are found. [ABSTRACT FROM AUTHOR]

We prove that in a Finsler manifold with vanishing χ -curvature (in particular with constant flag curvature) some non-Riemannian geometric structures are geodesically invariant and hence they induce a set of non-Riemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald curvature, or as functions of the mean Cartan torsion and the mean Landsberg curvature. [ABSTRACT FROM AUTHOR]

Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we provide a practical and easy way of computing these first integrals as the coefficients of a characteristic polynomial. [ABSTRACT FROM AUTHOR]

The feedback integrators method is improved, via the celebrated Dirac formula, to integrate the equations of motion for mechanical systems with holonomic constraints so as to produce numerical trajectories that remain in the constraint set and preserve the values of quantities, such as energy, that are theoretically known to be conserved. A feedback integrator is concretely implemented in conjunction with the first-order Euler scheme on the spherical pendulum system and its excellent performance is demonstrated in comparison with the RATTLE method, the Lie–Trotter splitting method, and the Strang splitting method. [ABSTRACT FROM AUTHOR]

In this paper, we study the limit cycles in the discontinuous piecewise linear planar systems separated by a nonregular line and formed by linear Hamiltonian vector fields without equilibria. Motivated by [Llibre & Teixeira, 2017], where an open problem was posed: Can piecewise linear differential systems without equilibria produce limit cycles? We prove that such systems have at most two limit cycles, and the limit cycles must intersect the nonregular separation line in two or four points. More precisely, the exact upper bound of crossing limit cycles is two, and this upper bound can indeed be reached: either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points. Based on Poincaré map, the stability of various limit cycles is also proved. In addition, we give some concrete examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus. [ABSTRACT FROM AUTHOR]

Motivated by the definition of rigid centres for planar differential systems, we introduce the study of rigid centres on the center manifolds of differential systems on $\mathbb {R}^{3}$. On the plane, these centres have been extensively studied and several interesting results have been obtained. We present results that characterize the rigid systems on $\mathbb {R}^{3}$ and solve the centre-focus problem for several families of rigid systems. [ABSTRACT FROM AUTHOR]

The Nosé–Hoover system is a basic primitive model for the molecular dynamics simulations, which describes the equilibrium characterized by canonical distributions at a constant temperature. Its simplest form, called the Nosé–Hoover oscillator (NH), is a three-dimensional quadratic polynomial system that admits both regular invariant tori and chaotic trajectories. Recently, a similar system, called the generalized Nosé–Hoover oscillator (GNH), was constructed to show the coexistence of invariant tori and topological horseshoe for time-reversible systems in ℝ 3 . This paper aims to study the integrability of both NH and GNH models. We show that (i) in the case of α = 0 , both NH and GNH models are integrable by quadratures and the general solutions are given; (ii) in the case of α ≠ 0 the non-existence of either global analytic first integrals or Darboux first integrals of two models are discussed and a complete characterization of Darboux polynomials and exponential factors are given; (iii) both NH and GNH models are not rationally integrable in an extended Liouville sense except for several parameter values by an extended Morales–Ramis theory; (iv) the reduced systems of NH and GNH models at the Poincaré compactification balls are integrable, which yields complete descriptions of dynamics at infinity for NH and GNH models. Interestingly, the topological structure of NH and GNH models at infinity strongly depends on the sign of α. These results may help us better understand the complex and rich dynamics of nonlinear time-reversible systems. [ABSTRACT FROM AUTHOR]