The global phase portraits (GPPs) of Hamiltonian systems composed of a linear part plus homogeneous nonlinear terms have been extensively studied. However, most results are confined to the systems where the nonlinear terms are of degree less than five, leaving the case of quintic and higher degrees largely unexplored. This paper investigates the complete classification of GPPs in reversible equivariant Hamiltonian systems of linear plus quintic homogeneous polynomials. Firstly, by analyzing the distribution of zero coefficients in the nonlinear part and utilizing the invariance of the GPPs under coordinate axis swaps, the studied system is classified into four distinct categories, for each of which a normal form is provided. Secondly, the finite and infinite singular point distributions and types of these normal forms are analyzed. For the analysis at infinity, we employ a symmetry-reduced compactification that takes advantage of the two-axis symmetry, allowing all infinite singular points to be examined through a single oblique local chart. Finally, ten distinct global topological structures are obtained on the Poincaré disk, and the corresponding bifurcation diagrams are presented.
This paper investigates the number of limit cycles in a hyperelliptic gourd-shaped Hamiltonian system under polynomial perturbations. Specifically, we consider the perturbed system ẋ=y, ẏ=x-5/4x^3+1/4x^5+ε f(x,δ )y, where 0<|ε | ≪ 1 and f(x,δ )=∑ _i=0^6 a_i x^i is a polynomial of degree six. By studying limit cycles bifurcating near the elementary centers, the double homoclinic loop and the heteroclinic loop, we obtain lower bounds for the number of limit cycles and their corresponding configurations. Furthermore, under the assumption that some parameters a_i vanish, we derive sharp upper bounds for the number of limit cycles by constructing ECT-systems and using geometric methods.
This paper deals with the limit cycle bifurcation problem of a cubic Hamiltonian system with a global isochronous center under piecewise smooth small perturbations. This system contains two perturbation parameters with different scales. By using the Melnikov function, we obtain that the two-parameter perturbation can produce up to two more crossing limit cycles than the single-parameter perturbation.
This paper focuses on the (2+1)-dimensional Maccari system with multiplicative Brownian noise, and conducts systematic research on the theoretical method, solution construction, nonlinear dynamics and numerical verification. To address the lack of diversity in solitary wave solutions, nonlinear dynamical analysis and high-precision numerical verification of the stochastic system, and to establish a stochastic wave model consistent with the actual physical environments. The Jacobi Elliptic Functions (JEFs) method is adopted to solve the stochastic nonlinear partial differential system rather than the traditional extended tanh method, and the partial differential system is simplified to a system of ordinary differential equations through a traveling wave transformation. Nonlinear dynamical analysis is carried out by combining phase space trajectories, power spectral density and Lyapunov exponents. The adaptive Runge–Kutta method is used to numerically verify the Hamiltonian system. The JEFs method can generate a large number of double-periodic elliptic function wave solutions, and construct various solitary wave solutions including parabolic, needle-shaped, periodic, bright and single breather waves. The noise intensity, wave velocity, and medium characteristic parameters can effectively regulate the evolution of waveform, which is consistent with real physical multi-factor coupling scenarios. Dynamical analysis reveals the topological structure and stability of equilibrium points, and the system shows no chaotic behavior when not subjected to external excitation, whereas chaotic behavior emerges when the system is under external forcing within the investigated parameter range. The numerical solution achieves high accuracy and strictly satisfies the energy conservation property of the Hamiltonian system. This work establishes a complete research framework covering methodological innovation, solution construction and numerical verification, which breaks through the limitation of finite solution types in traditional methods for stochastic nonlinear wave equations. This study provides theoretical reference and numerical support for the analysis and application of stochastic nonlinear waves in plasma physics, nonlinear optics and fluid dynamics.
Let X={X_k}_k=0^∞ be a sequence of compact metric spaces X_k and T={T_k:X_k→ X_k+1}_k=0^∞ a sequence of continuous mappings. T_k:X_k→ X_k+1 . The pair (X,T) is called a nonautonomous dynamical system. In this paper, we introduce a variety of topological pressures ( Q , Q , P , P , P^L , P^U , P^B and P^P ) for potentials f={f_k∈ C(X_k,ℝ)}_k=0^∞ on subsets Z ⊂ X_0 which are parallel to those concepts of fractal dimensions, and we provide various key properties that are crucial to the study of nonautonomous dynamical systems. Especially, we obtain the power rules and product rules of these pressures, and we also show they are invariants under equiconjugacies of nonautonomous dynamical systems and equicontinuity on f .
In this study, we investigate representative explicit parametric traveling-wave solutions of a ϕ ^8 field model and a highly nonlinear Schrödinger equation. Using a dynamical systems approach, we describe representative phase-portrait bifurcations for selected parameter regimes. By analyzing selected homoclinic, heteroclinic, and periodic orbits, we derive explicit parametric representations that yield solitary, kink, and periodic wave solutions.
We develop and analyse a theory for thermal convection in a Darcy porous material taking into account the concept of local thermal non-equilibrium whereby the temperature in the solid skeleton may be different from that in the saturating fluid. The thermal microstructure of the solid skeleton is such that we adopt a Jeffreys theory for heat conduction in this part of the continuous medium. The Jeffreys theory in the solid has a mathematical form analogous to that proposed by Sir Harold Jeffreys, F.R.S., for a fluid, although now rather than being a relation between stress and the symmetric part of the velocity gradient as in the original case, the relation is between the heat flux and the temperature gradient. A novel effect is that oscillatory convection requires the Maxwell coefficient to exceed the Jeffreys coefficient, while the precise stationary–oscillatory transition depends separately on both coefficients and on the strength of thermal interaction between the solid and fluid phases.
This paper investigates a variable coefficient (2+1)-dimensional shallow water wave equation describing wave-propagation dynamics in inhomogeneous media. First, Painlevé analysis is carried out using the WTC-Kruskal algorithm, and the corresponding integrability conditions of the equation are established. Subsequently, based on the Hirota bilinear method, we derive the bilinear form and bilinear auto-Bäcklund transformations of the equation. Within this framework, multiple soliton solutions are systematically constructed, and breather solutions and lump solutions are obtained. The degeneration from breather to lump solutions is clarified. In addition, hybrid localized solutions describing soliton-breather, soliton-lump, and breather-lump interactions are constructed. Finally, the linear stability of the equation is further analyzed. The influence of different time-dependent coefficients on the propagation characteristics of these nonlinear waves is investigated analytically and graphically. The results show that different coefficient profiles can modulate the dynamical behaviors of the wave structures, leading to straight, parabolic, S-shaped, and periodic evolution patterns.
The goal of this article is to determine within the theory of polynomial differential systems the role of the geometry of their invariant algebraic curves and in particular of their multiplicities as well as of the multiplicities of singularities of the systems located on them, in their Liouvillian integrability. By analyzing in depth the family of Liouvillian integrable systems QSL_≥ 4 of quadratic differential systems with real coefficients and with invariant lines of total multiplicity at least 4 we determine this role. We also show that along with the geometry and in particular the multiplicities of the curves and of singularities, also Diophantine equations play an important role. Our theorems give specific reasons for why the class splits in three subclasses with distinct kinds of integrating factors, depending on how the systems can be perturbed into generic ones, i.e. systems with all their invariant algebraic curves as well as their singularities simple (of multiplicity one). Analyzing the limiting process for integrating factors of the perturbed systems as the perturbation parameter ϵ tends to zero we get to understand why we end up with three types of integrating factors.
The traveling wave system of the Nesterenko equation is a singular nonlinear ordinary differential equation. By using the method of dynamical systems, we investigate bifurcations of phase portraits and the dynamical behavior of solutions for this singular traveling wave system (for n=∓ 2,± 3). The exact explicit parametric representations of some bounded traveling wave solutions are derived in the given parameter regions. More than 20 explicit exact parametric representations of the periodic solutions, periodic peakon, peakon, solitary wave solutions as well as compacton solutions are given.
In this paper, we study a piecewise-smooth Bazykin predator–prey model with the weak Allee effect and Holling type II functional response. The model is a regular-singular system with a natural slow-fast structure caused by the small intraspecific competition parameter of predators. By constructing a Poincaré map and using qualitative analysis together with geometric singular perturbation theory, we show that the system exhibits rich and novel dynamical behaviors. The system has a standard Hopf limit cycle and various types of crossing limit cycles, including relaxation oscillations as well as both small-amplitude and large-amplitude limit cycles associated with a boundary equilibrium. Furthermore, we demonstrate that under three distinct parameter regimes, the system can display complex dynamical phenomena: a heteroclinic orbit involving three equilibria; the coexistence of a homoclinic orbit and a family of heteroclinic orbits connecting two equilibria; and the coexistence of a stable Hopf limit cycle with both an unstable limit cycle and a stable relaxation oscillation. This work provides a theoretical foundation for explaining nonlinear ecological phenomena such as abrupt population shifts and delayed responses. In addition, numerical simulations validate the theoretical results and clarify their ecological implications.
This paper investigates optical soliton solutions and bifurcation in the generalised resonant nonlinear Schrödinger equation (NSE). The generalised resonant NSE portrays wave propagation in optical fibers. The modified F-expansion technique (MFT) is utilized to construct solitary, periodic, singular, dark, and kink-type wave profiles. By employing a travelling wave transformation, the generalised resonant NSE is simplified to a singular dynamical system, which is subsequently converted into a regular dynamical system via variable substitution. The results show that the first integrals of both the singular and regular systems are mathematically equivalent, ensuring topological consistency between their trajectories. This equivalence facilitates a comprehensive examination of their phase portraits, highlighting their topological and geometric characteristics. Graphical representations in 2D and 3D of the solutions are provided to demonstrate their features physically, with parameters selected to emphasize the impacts of the generalised resonant NSE. These findings elucidate the kinetics of nonlinear wave propagation in optical systems.
If we want to classify a large number of objects, we normally have to divide the set into a few subsets and then classify these subsets one by one. In the study of phase portraits of quadratic systems, different divisions have been tried and some subsets of each division have been classified. So far, no division has been efficient enough to allow a complete classification. In this article we present the most advanced study based on one of these divisions. Specifically, we study a division proposed in [7] which divides quadratic systems into ten different normal forms with six parameters. We will provide the complete classification of phase portraits for six of these normal forms. Altogether with [1], in which a seventh normal form has been studied, we bring together all the current knowledge on this division. Unfortunately the three remaining normal forms are not easy to reduce to simpler forms that allow their complete study.
This paper focuses on the study of Kukles systems of degree m > 1 . Using the theory of Melnikov functions, we estimate the maximum number of limit cycles surrounding the origin of the systems, according to whether the origin is a focus, a node, a saddle, or a nilpotent singularity. In particular, we present concrete examples of phase portraits for m = 3 .
In this paper we investigate the planar circular restricted ( 3 + 1 )-body problem. A massless particle moves under the Newtonian gravitational influence of three primaries. These primaries revolve in circular orbits around their common center of mass within the same plane, forming a collinear configuration. We establish the existence and nonlinear stability of equilibrium points under the condition that two of the primaries have equal masses, denoted by μ . Through analytical and numerical methods, we identify a critical mass parameter μ _1 ≈ 0.0743 where a Hamiltonian–Hopf bifurcation occurs at non-collinear equilibrium points. Furthermore, we prove the existence of Hill-type periodic orbits and KAM tori surrounding these periodic orbits in this circular restricted four-body problem.
In this paper, we investigate the distributional n -chaos of the functional envelope system (S(Σ ), F_T) induced by a nontrivial weakly mixing dynamical system (Σ , T) with the shadowing property, where Σ is an arbitrary closed subset of A^ℕ and T:Σ→Σ is a continuous map. We prove that there exists a dense Mycielski set K ⊂ S(Σ ) such that any n pairwise distinct points in K form a distributional n - d_n -scrambled tuple, where n ≥ 2 and d_n>0 . In particular, it follows that (S(Σ ), F_T) is a distributional n - d_n -chaotic system with a dense Mycielski scrambled set.
A variable-coefficient bilinear neural network method is proposed for deriving analytical solutions to the variable-coefficient Kadomtsev–Petviashvili equation and the (2+1)-dimensional variable-coefficient Sawada–Kotera equation. By constructing adaptive 3-2-2-1 and 3-3-2-1 neural network models, we present abundant analytical solutions for these equations. Additionally, a variable-coefficient positive quadratic function method is introduced to obtain abundant lump-type solutions for the Sawada–Kotera equation. Dynamic properties of the derived solutions, including amplitude evolution and spatial interactions, are visualized through three-dimensional surface plots and density graphics, revealing their nonlinear wave behaviors.
We study the delay differential equation x'(t) = a [x(t) - x(t - 1)] - g(x(t - τ )) where a>0 , τ >0 , and g:ℝ∋ u↦ u |u|^κ∈ℝ with κ >0 . This equation is a modification of the Brunovský–Erdélyi–Walther price model by incorporating a reaction delay τ >0 . For any a>0 and κ >0 , by the Kaplan–Yorke method and the homogeneity of the nonlinear function g, we find a countable and dense set of delays τ in (0,∞ ) for which there exists a periodic solution. A consequence is that global asymptotic stability of the zero solution cannot be expected if a∈ (0,1) , in contrast to the case τ =0 . For a∈ (0,1) and τ∈ (0,1] , an R_1>0 is constructed such that 0 attracts the ball of radius R_1 with center at 0. Local asymptotic stability of the zero solution follows as well. It is also shown that, for any a∈ (0,1) , the region of attraction of 0 tends to the whole phase space C([-1,0],ℝ) as τ→ 0^+.
In this paper, we study limit cycle bifurcations near a symmetric double homoclinic loop in planar cubic near-Hamiltonian systems by high-order analysis. We find 6 limit cycles from the exterior period annulus near the loop by the third-order Melnikov function, where a type of pseudo-Abelian integrals are involved. Furthermore, using Melnikov functions of different orders for the exterior and inner annuli respectively, we find a new distribution (6, 1, 1) for limit cycles bifurcating near the loop.
We investigate resonant Hamiltonians with n degrees of freedom to which we attach a small perturbation of polynomial type. Our analysis is based on singular reduction theory within the setting of Hamiltonian dynamics. In order to perform symplectic reduction, the system is first transformed into normal form, after which the higher-order terms of the transformation are truncated. This procedure results in a reduced Hamiltonian system defined on the corresponding orbit space, which is a manifold in the case of regular reduction and an orbifold in the case of singular reduction. We then classify the singularities of the orbit space. Continuous families of periodic solutions of the original system are characterized through the study of nondegenerate critical points on the reduced space. The local desingularization of all types of orbifold singularities is achieved by means of symplectic transformations. This framework provides an effective method for approximating the characteristic multipliers of the periodic solutions, obtained by computing the eigenvalues of the linearized reduced Hamiltonian at the critical point associated with the corresponding periodic orbit. Several examples with two, three, and four degrees of freedom are presented to illustrate the theoretical results.
