Two-dimensional Autonomous System Research Articles

Integration of a degenerate system of ODEs

The integrability of a two-dimensional autonomous polynomial system of ordinary differential equations (ODEs) with a degenerate singular point at the origin that depends on six parameters is investigated. The integrability condition for the first quasihomogeneous approximation allows one of these parameters to be fixed on a countable set of values. The further analysis is carried out for this value and five free parameters. Using the power geometry method, the system is reduced to a non-degenerate form through the blowup process. Then, the necessary conditions for its local integrability are calculated using the method of normal forms. In other words, the conditions for the parameters under which the original system is locally integrable near the degenerate stationary point are found. By resolving these conditions, we find seven twoparameter families in the five-dimensional parametric space. For parameter values from these families, the first integrals of the system are found. The cumbersome calculations that occur in the problem under consideration are carried out using computer algebra.

Propagation characteristics of nonlinear dust acoustic solitary waves in complex plasma with nonthermal electrons and trapped ions

The propagation characteristics of nonlinear dust acoustic solitary waves in a complex plasma system with nonthermal electrons and trapped ions are investigate in this work. The nonlinear dispersion relation of dust acoustic waves is obtained by using the linear method, and the two-dimensional autonomous system governing the motion of nonlinear dust acoustic waves is derived by using the Sagdeev potential method. At the same time, the specific expression of the Sagdeev potential function is obtained based on the Sagdeev potential equation. The numerical simulations are used to analyze the phase portraits of the two-dimensional autonomous system, revealing the linear periodic wave orbits, nonlinear periodic wave orbits, and homoclinic orbits co-existing in the complex dusty plasma system with nonthermal electrons and trapped ions. Furthermore, from the variations of the Sagdeev potential function with different system parameters it follows that only the compressive solitary waves exist in this complex plasma system. The significant influences of various system parameters on the amplitude, width, and waveform of the nonlinear dust acoustic solitary wave in the complex plasma system are discussed in detail. The results demonstrate that the Mach number, the nonthermal electrons and trapped ions, undisturbed dust particle number density, temperature, and charge have important effects on the propagating characteristics of the nonlinear dust acoustic solitary waves in a complex plasma with nonthermal electrons and trapped ions.

Center conditions to find certain degenerate centers with characteristic directions

We consider the two-dimensional autonomous systems of differential equations where the origin is a monodromic degenerate singular point, i.e., with null linear part. In this work we give two heuristic procedures to obtain some center conditions (perhaps not necessary) for certain degenerate centers at the origin although they have characteristic directions.

Dynamics Analysis of Cannibalistic Model With Density Dependence in Mature Stage

Based on considering mature individuals with density dependence, a two-stage structure model of egg-mature individuals with cannibalism is established. The dynamic behavior of the equilibrium of the model is discussed from two aspects: when no cannibalism exists, the global asymptotic stability of the equilibria is proved by the Lyapunov theorem; when cannibalism exists, it makes the model have saddle-node bifurcation by using the center manifold theorem. By constructing Dulac function, there is no limit cycle in the two-dimensional autonomous system, therefore, the global stability of the equilibria is obtained. Finally, the theoretical results are verified by numerical simulation.

Propagating characteristics of nonlinear dust acoustic solitary waves in multicomponent dusty plasma

Nowadays, the dusty plasma has become an interesting new branch of the plasma physics. As is well known, the dusty plasmas play a significant role in the space, astrophysical and laboratory environments. In these days, the studying of the nonlinear waves in dusty plasma has attracted researchers’ attention, in order to explain many basic phenomena in the plasma physics. The nonlinear waves play an important role in studying dusty plasma environments, such as the aster-oid zones, the earth’s mesosphere, and the planetary rings. In this work, the propagating characteristics of nonlinear dust acoustic solitary waves in a multicomponent dusty plasma which is composed of positively charged dust particles, streaming protons and electrons, Kappa distributed electrons and ions are studied in detail. The Sagdeev potential method is employed to investigate the large amplitude dust acoustic waves. It has an evidence for the existence of compressive and rarefractive solitary waves. With the help of the Sagdeev potential method, the Sagdeev potential function and the bifurcation analysis of phase-portrait are obtained. Firstly, the Sagdeev potential function is obtained by the Sagdeev potential method. Then, the variations of phase diagram with different parameters in a two-dimensional autonomous system in the multicomponent dusty plasma system are investigated. It is found that the system has the linear wave solutions, nonlinear wave solutions, and solitary wave solutions at the same time. Meanwhile, the existence of different wave behaviors is closely related to various system factors. Moreover, it is found that only the rarefractive solitary waves exist in the multicomponent dusty plasma system by using the numerical simulation technique. Finally, the important influence of system parameter on the phase diagram, the Sagdeev potential function and the propagating characteristics of nonlinear dust acoustic solitary waves are discussed clearly. The results show that the different system parameters such as Mach number <i>M</i>, the masses, the temperatures, the number densities, the charge numbers of multiple particles and the Kappa distribution parameters for ions and electrons have important effects on the amplitudes, the widths and the waveforms of nonlinear dust acoustic solitary waves.

Trajectories and Singular Points of Two-Dimensional Fractional-Order Autonomous Systems

In this paper, we study the trajectories and singular points of two-dimensional fractional-order planar autonomous linear system involving the Caputo-Fabrizio fractional derivative. By the corresponding fractional integral of the Caputo-Fabrizio fractional derivative, we obtain the analytical solutions for the fractional-order planar autonomous linear system, and then, we discuss the behavior of the trajectories for the mentioned autonomous linear system. Furthermore, we consider the existence of singular points in the trajectories. We discuss the conditions under which the singular point is stable or unstable. By determining the value range of the parameters, we obtain the theorems on the type of singular points. Finally, some examples are given to verify the analysis for the mentioned autonomous linear system.

(3 + 1) dimensional nonlinear ion acoustic waves in multicomponent plasma containing Kappa distributed electrons

The (3 + 1) dimensional nonlinear ion acoustic waves in multicomponent complex plasma with Kappa electron distribution are studied in this work. The Zakharov-Kuznetsov (ZK) equation for ion acoustic wave is investigated by the reductive perturbation method. The variation of nonlinear ion acoustic wave with system parameter is obtained. Meanwhile, the Sagdeev potential function is obtained by using the Sagdeev potential method. And the phase diagram in a two-dimensional autonomous system of the multicomponent complex plasma with Kappa electron distribution is studied. Finally, the important influence of system parameter on the phase diagram, the Sagdeev potential function and the propagating characteristics of (3 + 1) dimensional nonlinear acoustic solitary waves are discussed in detail.

Generalized model of interacting dark energy and dark matter: Phase portrait analysis for evolving universe

The main aim of this work is to give a suitable explanation of present accelerating universe through an acceptable interactive dynamical cosmological model. A three-fluid cosmological model is introduced in the background of Friedmann–Lemaître–Robertson-Walker asymptotically flat spacetime. This model consists of interactive dark matter and dark energy with baryonic matter, taken as perfect fluid, satisfying barotropic equation of state. We consider dust as the candidate of dark matter. A scalar field [Formula: see text] represents dark energy with potential [Formula: see text]. Einstein’s field equations are utilized to construct a three-dimensional interactive autonomous system by choosing suitable interaction between dark energy and dark matter. We take the interaction kernel as [Formula: see text], where [Formula: see text] indicates the density of dark energy, [Formula: see text] is the interacting constant and [Formula: see text] is Hubble parameter. In order to explain the stability of this system, we obtain some suitable critical points. We analyze stability of obtained critical points to show the different phases of universe and cosmological implications. Surprisingly, we find some stable critical points which represent late-time dark energy-dominated era when a model parameter [Formula: see text] is equal to [Formula: see text]. We introduce a two-dimensional interactive autonomous system and after phase portrait analysis of it, we get several stable points which represent dark energy-dominated era and late-time cosmic acceleration simultaneously. Here, we also demonstrate the variation in interaction at vicinity of phantom barrier [Formula: see text]. From our work, we can also predict the future phase evolution of the universe.

The Curvature Property of a Linear Dynamical System

In this work a two-dimensional smooth autonomous dynamical system is regarded as a three-dimensional Riemannian manifold and it is shown that the scalar curvature of a linear dynamical system $\text{d}x/\text{d}t=ax+by$, $\text{d}y/\text{d}t=cx+dy$ is non-positive. The manifold is scalar-flat iff $b=-c$ and $a=d=0$.

Exact Stability and Instability Regions for Two-Dimensional Linear Autonomous Multi-Order Systems of Fractional-Order Differential Equations

Abstract Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterised, in terms of the main diagonal elements of the systems’ matrix, as well as its determinant.

Mixed-Mode Oscillations Based on Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model.

Abstract This article highlights particular mixed-mode oscillations (MMO) based on canard explosion observed in a fractional-order Fitzhugh-Nagumo (FFHN) model. In order to rigorously analyze the dynamics of the FFHN model, a recently introduced mathematical notion, the Hopf-like bifurcation (HLB), which provides a precise definition for the change between a fixed point and an S−asymptotically T−periodic solution, is used. The existence of HLB in this FFHN model is proved and the appearance of MMO based on canard explosion in the neighborhoods of such HLB points are numerically investigated using a new algorithm: the global-local canard explosion search algorithm. This MMO is constituted of various patterns of solutions with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied simultaneously. On the basis of such numerical experiment, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional-order close to one. Therefore, this very simple two-dimensional FFHN model, presents an incredible ability to mimic the complex dynamics of neurons.

Study on the generation mechanism of bright and dark solitary waves and rogue wave for a fourth-order dispersive nonlinear Schrödinger equation

In this paper, we study the generation mechanism of bright and dark solitary waves and rogue wave for the fourth-order dispersive nonlinear Schrödinger (FODNLS) equation, which can not only model the nonlinear propagation and interaction of ultrashort pulses in the high-speed optical fiber transmission system, but also govern the nonlinear spin excitations in the onedimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole-dipole interaction. Firstly, via the phase plane analysis, we obtain both the homoclinic and heteroclinic orbits for the two-dimensional plane autonomous system reduced from the FODNLS equation. Further, we derive the bright and dark solitary wave solutions under the corresponding conditions, which reveals the relationship between the homoclinic (heteroclinic) orbit and solitary wave. Secondly, based on the exact first-order breather solution of the FODNLS equation over a nonvanishing background, we give the explicit expressions of group and phase velocities, and reveal that there exists a jump in both the velocities. Finally, in order to verify that the breather becomes a rogue wave at the jumping point, we obtain the first-order rogue wave solution by taking the limit of the breather solution at such point, which confirms the relationship of the generation of rogue wave with the velocity discontinuity.

Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

Chaotic and non-chaotic response to quasiperiodic forcing: limits to predictability of ice ages paced by Milankovitch forcing

It is well known that periodic forcing of a nonlinear system, even of a two-dimensional autonomous system, can produce chaotic responses with sensitive dependence on initial conditions if the forcing induces sufficient stretching and folding of the phase space. Quasiperiodic forcing can similarly produce chaotic responses, where the transition to chaos on changing a parameter can bring the system into regions of strange non-chaotic behaviour. Although it is generally acknowledged that the timings of Pleistocene ice ages are at least partly due to Milankovitch forcing (which may be approximated as quasiperiodic, with energy concentrated near a small number of frequencies), the precise details of what can be inferred about the timings of glaciations and deglaciations from the forcing is still unclear. In this paper, we perform a quantitative comparison of the response of several low-order nonlinear conceptual models for these ice ages to various types of quasiperiodic forcing. By computing largest Lyapunov exponents and mean periods, we demonstrate that many models can have a chaotic response to quasiperiodic forcing for a range of forcing amplitudes, even though some of the simplest conceptual models do not. These results suggest that pacing of ice ages to forcing may have only limited determinism.

Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris–Lecar neuronal model

Necessary and sufficient conditions are given for the asymptotic stability and instability of a two-dimensional incommensurate order autonomous linear system, which consists of a differential equation with a Caputo-type fractional-order derivative and a classical first-order differential equation. These conditions are expressed in terms of the elements of the system’s matrix, as well as of the fractional order of the Caputo derivative. In this setting, we obtain a generalization of the well-known Routh–Hurwitz conditions. These theoretical results are then applied to the analysis of a two-dimensional fractional-order Morris–Lecar neuronal model, focusing on stability and instability properties. This fractional-order model is built up taking into account the dimensional consistency of the resulting system of differential equations. The occurrence of Hopf bifurcations is also discussed. Numerical simulations exemplify the theoretical results, revealing rich spiking behavior. The obtained results are also compared to similar ones obtained for the classical integer-order Morris–Lecar neuronal model.

A general method for exploring three-dimensional chaotic attractors with complicated topological structure based on the two-dimensional local vector field around equilibriums

This paper proposed a new method for exploring three-dimensional chaotic attractors with complicated topological structure. Based on the index theory, the theoretical foundation of the new method is explained clearly and concisely, and the feasibility of this method is progressively illustrated by a multi-wing three-dimensional chaotic attractor derived from a two-dimensional continuous autonomous dynamical system with thirteen equilibrium points. Moreover, the experimental results are also discussed. To validate the generalization ability of the proposed method, another two chaotic systems are constructed based on the proposed method. The numerical results show the effectiveness of the proposed method.

Visual exploration of 2D autonomous dynamical systems

In an introductory course on dynamical systems or Hamiltonian mechanics, vector field diagrams are a central tool to show a system’s qualitative behaviour in a certain domain. Because of their low sampling rates and the involved issues of vector normalization, these plots give only a coarse insight and are unable to convey the vector field behaviour at locations with high variation, in particular in the neighbourhood of critical points. Similarly, automatic generation of phase portraits based on traditional sampling cannot precisely capture separatrices or limit cylces. In this paper, we present ASysViewer, an application for the interactive visual exploration of two-dimensional autonomous dynamical systems, using line integral convolution techniques for visualization, and grid-based techniques to extract critical points and separatrices. ASysViewer is addressed to undergraduate students during their first course in dynamical systems or Hamiltonian mechanics.

Dynamical behaviors and soliton solutions of a generalized higher-order nonlinear Schrödinger equation in optical fibers

Under study in this paper is a generalized higher-order nonlinear Schrodinger (GHNLS) equation with the third-order dispersion (TOD), self-steeping (SS) and stimulated Raman scattering effects , which describes the propagation of ultrashort pulses in optical fibers. Via the phase plane analysis, both the homoclinic and heteroclinic orbits are found in the two-dimensional plane autonomous system reduced from the GHNLS equation, which proves the existence of bright and dark soliton solutions from the viewpoint of nonlinear dynamics. Furthermore, through the method of binary Bell polynomials and auxiliary function, the explicit bright and dark soliton solutions under certain conditions are obtained. Particular analysis is made to study the effects of the higher-order on the double-hump bright and double-hole dark solitons. The results show that the self-phase modulation and SS parameters determine the interval between two humps for the double-hump bright soliton, while the one for the double-hole dark soliton is related with the TOD and SS effects. Moreover, numerical simulations show that the double-hump bright soliton and the double-hole dark soliton are more stable when the amplitude or depth is comparably small.

Global Attractor of Solutions of a Rational System in the Plane

We consider a two-dimensional autonomous system of rational difference equations with three positive parameters. It was conjectured that every positive solution of this system converges to a finite limit. Here we confirm this conjecture, subject to an additional assumption.

New exact wave solutions for Hirota equation

In this paper, we construct the topological or dark solitons of Hirota equation by using the first integral method. This approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solution is constructed through the established first integrals. This method is a powerful tool for searching exact travelling solutions of nonlinear partial differential equations (NPDEs) in mathematical physics.