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With both analytical and numerical methods, chaotic motions of the Rayleigh–Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation are investigated in this paper. Chaos arising from homoclinic or heteroclinic intersections is studied with the Melnikov method. Chaos threshold is obtained and chaotic feature on the system parameters is investigated in detail. Some new interesting dynamic phenomena including “controllable frequency interval”, “chaotic band” and “uncontrollable parameters” are presented and proved rigorously. Numerical simulations are given to verify chaos threshold obtained by the analytical results.

In this work a homoclinic-like loop of a 3D piecewise smooth vector field passing through a typical singularity is analyzed. We have shown that such a loop is robust in one-parameter families of Filippov systems. The basin of attraction of this connection is computed as well as its bifurcation diagram. It is worthwhile to mention that this phenomenon has no counterpart in the smooth world. Our approach relies on the analysis of first return maps and the technique used in this scenario is much more complex than the usual analysis of Poincare maps in planar Filippov systems.

The transfer between libration orbits in the Mars–Phobos system is challenging under the strong gravitational perturbation of Mars and Phobos. In this paper, the outer branches of manifolds of libration orbits around Phobos are employed for impulse and continuous thrust transfer between orbits, respectively. The 1:1 resonant orbits are found to be appropriate intermediate orbits for impulse transfer, which can connect the manifolds of various kinds of libration orbits. For the continuous thrust transfer, the convex optimization method is adopted to design the low-thrust trajectory in the three-body problem. The Poincare section is established to find the best matching points on the outer manifolds. The L 1 / L 2 homoclinic and L 1 − L 2 heteroclinic transfer trajectories are designed and discussed. The simulations show the feasibility of the transfer by outer manifolds with both impulse and continuous thrust. The proposed method can avoid the potential hazard caused by the complicated dynamics near Phobos and provide a reliable transfer method, which can be used in future Phobos exploration missions.

In this paper, we study a near-Hamiltonian system on the cylinder. First, we establish some general methods on the existence of limit cycles bifurcating from closed orbits of type II by the Melnikov function method, then we derive the expansions of the first order Melnikov function and consider the bifurcation problem of limit cycles near a double homoclinic loop. As an application, we discuss the number of limit cycles of a class of cylinder pendulum-like systems.

In this paper, bifurcation dynamics and infinite homoclinic orbits of (1:2)- asymmetric periodic motions in the Lorenz systems are studied through the discrete mapping method. The infinite homoclinic orbits pertaining to the unstable periodic motions on the bifurcation trees of (1:2)-asymmetric periodic motions to chaos are determined. The stability and bifurcations of periodic motions are determined through the eigenvalue analysis. A bifurcation tree of (1:2) asymmetric periodic motions, varying with the Rayleigh number, is presented by discrete nodes, and the corresponding harmonic frequency-amplitude characteristics of periodic motions on the bifurcation tree are discussed through finite Fourier series analysis. A sequence of period-doubling bifurcations is observed on the bifurcation tree of the (1:2)- asymmetric periodic motion to chaos. The scenario of (1:2)- asymmetric period-1, period-2, period-4 motions to chaos scenario is illustrated. Homoclinic orbits are as appearance or vanishing of unstable periodic motions on the bifurcation tree. Illustrations of periodic motions and homoclinic orbits are completed for intuitive demonstrations of the corresponding topological structures. This study extends the initial study of the bifurcation tree from the (1:1)-symmetric periodic motions to asymmetric periodic motions then to chaos in the Lorenz system, and the corresponding infinite homoclinic orbits induced by all unstable periodic motions can be determined. Such results can enrich the understanding of bifurcation dynamics of asymmetric periodic motions to chaos in the Lorenz system.

In this article, the bilinear form, Backlund transformation, Lax pair and infinite conservation laws of the (3+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation with time-dependent coefficients are constructed based on the Bell polynomials approach. N-soliton solutions are studied by means of introducing the complex conjugate condition technique and selecting appropriate test functions and parameters, including the hybrid solution of the a-order kink waves, b-order periodic-kink waves and c-order periodic-breather waves. The homoclinic test method is applied to investigate their dynamical interaction properties between different forms of hybrid-type solutions. Besides, a number of examples are presented by choosing different types of interactions among the hybrid-type solutions. Finally, we analyze the wave propagation direction and velocity to reflect the novel evolutionary behaviors in the three-dimensional profile of the model. These results are helpful to the study of local wave interactions in nonlinear mathematical physics.

This paper investigates the effects of dynamic loads and structural damping on the nonlinear behaviors of incompressible hyperelastic spherical shells modeled by the Yeoh strain energy function. Firstly, the dynamical modeling is formulated for the nonlinear behaviors of the shells by the variational principle, and the second order nonlinear ordinary differential equation describing the radially symmetric motion is obtained. Then, the dynamic behaviors, such as periodic, quasiperiodic and chaotic motions, are discussed under different loading types. Particularly, for constant loads, the first integral of the integrable Hamiltonian system without damping is given and it is numerically proved that there exists an asymmetric "∞" homoclinic orbit for the prescribed material parameters obtained in experiments; moreover, it is found that for different prestretches, the structural parameter has a completely different role on the nonlinearity of the system, and the basins of attraction are given with the structural damping. For periodic loads, there exist some interesting dynamic phenomena, i.e., quasiperiodic oscillation in the approximately integrable Hamiltonian system, limit cycles and chaos with the damping. The criterion for chaos is discussed by the Melnikov method combined with the numerical calculation and the chaos is further analyzed with the Poincare section and the phase plane.

When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or non-orientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case $\textbf{B}$, which is characterized by the fact that the codimension-two point gives rise to an additional homoclinic bifurcation, namely, a two-homoclinic orbit. To explain how the global manifolds organize phase space, we consider Sandstede's three-dimensional vector field model, which features inclination and orbit flip bifurcations. We compute global invariant manifolds and their intersection sets with a suitable sphere, by means of continuation of suitable two-point boundary problems, to understand their role as separatrices of basins of attracting periodic orbits. We show representative images in phase space and on the sphere, such that we can identify topological properties of the manifolds in the different regions of parameter space and at the homoclinic bifurcations involved. We find heteroclinic orbits between saddle periodic orbits and equilibria, which give rise to regions of infinitely many heteroclinic orbits. Additional equilibria exist in Sandstede's model and we compactify phase space to capture how equilibria may emerge from or escape to infinity. We present images of these bifurcation diagrams, where we outline different configurations of equilibria close to homoclinic flip bifurcations of case $\textbf{B}$; furthermore, we characterize the dynamics of Sandstede's model at infinity., 43 pages, 21 figures

We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward in understanding the generic cases of homoclinic flip bifurcations, which started with the study of the simpler cases \textbf{A} and \textbf{B}. In a three-dimensional vector field due to Sandstede, we compute relevant bifurcation curves in the two-parameter bifurcation diagram near the central codimension-two bifurcation in unprecedented detail. We present representative images of invariant manifolds, computed with a boundary value problem setup, both in phase space and as intersection sets with a suitable sphere. In this way, we are able to identify infinitely many cascades of homoclinic bifurcations that accumulate on specific codimension-one heteroclinic bifurcations between an equilibrium and various saddle periodic orbits. Our computations confirm what is known from theory but also show the existence of bifurcation phenomena that were not considered before. Specifically, we identify the boundaries of the Smale--horseshoe region in the parameter plane, one of which creates a strange attractor that resembles the R\"{o}ssler attractor. The computation of a winding number reveals a complicated overall bifurcation structure in the wider parameter plane that involves infinitely many further homoclinic flip bifurcations associated with so-called homoclinic bubbles., Comment: 42 pages, 23 figures

We develop methods giving sufficient conditions for the existence of bounded solutions near homoclinic orbits of planar R.D.E.s. We provide families of bounded solutions near the homoclinic by decomposing the corresponding variational equation of the system on the space of solutions. We apply the results for controlling a power-law oscillatory system with a retarded feedback control. Our results are numerically verifiable and do not meet the restrictions of classic methods (such as Melnikov’s integrals and Poincare’s map).

The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using Ambrosetti-Rabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.

The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

This paper is concerned with the chaotic dynamics of a rotating pendulum system with bistable characteristics subjected to a viscous damping and a harmonic forcing. As a prototype of the single-degree-of-freedom system with bistable characteristics, this pendulum system exhibits a transition from smooth to discontinuous dynamics by changing a geometrical parameter. The dynamic behaviors of the unperturbed system with irrational nonlinearity bear significant similarities to the coupling of a simple pendulum and the smooth and discontinuous (SD) oscillator with the coexistence of the standard homoclinic orbits of Duffing type and pendulum type and the coexistence of the nonstandard homoclinic orbits of SD type and pendulum type in the smooth and discontinuous case, respectively. For the perturbed smooth system, we present an approximate technique to analytically obtain the lower bound line for horseshoes chaos arising from the homoclinic orbits of Duffing-type and Pendulum-type tangling, which overcomes the natural difficulties of solving the analytical expression of the homoclinic orbits and calculating the complicated Melnikov integrals. The chaotic thresholds of the perturbed discontinuous system are calculated by applying the numerical technique due to its non-smooth feature. Numerical simulations are carried out to certify the chaotic thresholds, which show the efficiency of the proposed techniques and demonstrate the predicated chaotic motions. Finally, different types of chaotic motions are illustrated via the cylindrical phase portraits. The contribution of this study is also helpful for exploring the dynamical behaviors of the complex nonlinear dynamical system containing the standard homoclinic or heteroclinic orbit in terms of the quantitative calculation.

The aim here is to study complex dynamical behavior in a new kind of non-smooth systems with two discontinuous boundaries in space $$\mathbb {R}^3$$ . This paper provides an applicable method for accurately detecting coexistence of singular cycles, including (a) homoclinic and homoclinic cycles, (b) homoclinic and heteroclinic cycles, (c) heteroclinic and heteroclinic cycles. It is further proposed some criteria for guaranteeing three types of coexistence in the considered system. Moreover, existence conditions of chaos induced by such coexistence are established. Finally, three numerical examples are provided to verify the validity of theoretical results.

In a previous paper we studied parametrized autonomous systems and gave a computable criterion that an approximate orbit connecting hyperbolic equilibria is shadowed by a true connecting orbit. This criterion was used to give rigorously verified examples of Shilnikov saddle-focus homoclinic orbits in three dimensions. This involved verifying a condition on the eigenvalues of the linearization at the equilibrium. In dimensions greater than three, there are three more conditions which must be established: general position, asymptotic tangency and a transversality condition. In this paper we give computable criteria for verifying these three conditions. An example in four dimensions, in which detailed rigorous computations are carried out, is given.

In this paper, we study the chaos control of pendulum system with vibration of suspension axis for ultra-subharmonic resonance by using Melnikov methods, and give a necessary condition for controlling heteroclinic chaos and homoclinic chaos, respectively. We give some bifurcation diagrams by numerical simulations, which indicate that the chaos behaviors for ultra-subharmonic resonance may be inhibited to periodic orbits by adjusting phase-difference of parametric excitation, and prove that results obtained are very effective in inhibiting chaos for ultra-subharmonic resonance.

In this paper we consider the Degasperis-Procesi equation, which is an approximation to the incompressible Euler equation in shallow water regime. First we provide the existence of solitary wave solutions for the original DP equation and the general theory of geometric singular perturbation. Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Degasperis-Procesi equation is transformed into the slow-fast system by using the traveling wave transformation. It is proved that the perturbed equation also has a homoclinic orbit, which corresponds to a solitary wave solution of the delayed Degasperis-Procesi equation.

Seeking exact solutions of higher-dimensional nonlinear partial differential equations has recently received tremendous attention in mathematics and physics. In this paper, we investigate exact solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation which describes nonlinear wave propagation in incompressible fluid. Firstly, by means of extended homoclinic test approach, we get eight kinds of non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Then, combining the improved tanh function method and new ansatz solutions, we obtain abundant new exact non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. These results include not only many results obtained in other literatures, but also some new exact non-traveling wave solutions. Moreover, the exact kink wave solutions, periodic solitary wave solutions and singular solitary wave solutions are given when arbitrary functions contained in these solutions are taken as some special functions.

How to model non-smooth dynamical systems and study its non-smooth global dynamics by developing analytical methods is now an important topic. In this paper, a new bistable nonlinear oscillator with bilateral elastic constraints and a three-piecewise nonlinear restoring force is established to study the perturbation of viscous damping and an external harmonic excitation. A five-piecewise linear approach to the restoring force can transform the original dimensionless dynamical equations into a two-dimensional piecewise-defined Hamiltonian systems, separated by four switching manifolds with symmetrical characteristic and subjected to damping dissipation and a periodic excitation. The geometrical structure of the unperturbed system has a pair of symmetrical piecewise-defined homoclinic orbits and each of them transversally and continuously crosses two switching manifolds. The analytical Melnikov method is extended here to fit the theoretical framework for analyzing homoclinic bifurcations and chaotic dynamics for non-smooth oscillators with multiple switching manifolds. A global perturbation technique is presented to calculate the distance between the stable and unstable manifolds and derive the Melnikov function in the form of piecewise-integration. A major innovation here is no need to extend the vector filed near the switching manifolds and carry out rigorous perturbation analysis with geometrical intuition to derive the Hamilton energy differences of trajectories involving infinite time. The developed Melnikov function can be directly used to detect the parameter threshold of homoclinic chaos in the bistable nonlinear oscillator under bilateral elastic collision. Finally, the effectiveness of the Melnikov analysis for homoclinic chaos is verified by simulations.

Variable-coefficient nonlinear evolution equations offer us with more real aspects in the inhomogeneities of media and non-uniformities of boundaries than their constant coefficients in some real-world problems. A $$(3+1)$$ -dimensional variable coefficient Date–Jimbo–Kashiwara–Miwa (vcDJKM) equation and a $$(3+1)$$ -dimensional variable coefficient Boiti–Leon–Manna–Pempinelli (vcBLMP) equation are studied based on the homoclinic test method and construction of abundant solutions for two kinds of equations are obtained with the help of symbolic computation. In order to study the interaction superposition solutions with different functions, an existence theorem and corollary about superposition solutions of the $$(3+1)$$ -dimensional vcBLMP equation are proved. Via three-dimensional profiles with the help of mathematics, the propagation and the dynamical behavior of these solutions are analyzed by choosing different arbitrary variable coefficients. Compare with the published studied, some completely new results are presented in this paper.

We study the existence of homoclinic solutions for a class of non-periodic discrete nonlinear Schrodinger equations, where nonlinearities are super-linear at infinity, and primitive functions of nonlinearities are allowed to be sign-changing. By using some weaker conditions, our result extends and improves some results in the literature. Besides, we also give examples to illuminate our results.

In this paper, we investigate non-traveling wave solutions of the (3+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa (VC-DJKM) equation, which describes the real physical phenomena owing to the inhomogeneities of media. By combining the extended homoclinic test approach with variable separation method, we obtain abundant new exact non-traveling wave solutions of the (3+1)-dimensional VC-DJKM equation. These results with a parabolic tail or linear tail reveal the complex structure of the solutions for (3+1)-dimensional VC-DJKM equation. Moreover, the tail in these solutions maybe give a prediction of physical phenomenon. When arbitrary functions contained in these non-traveling wave solutions are taken as some special functions, we can get the kink-type solitons, singular solitary wave solutions, and periodic solitary wave solutions, and so on. As the special cases of our work, the corresponding results of (3+1)-dimensional DJKM equation, (2+1)-dimensional DJKM equation, (2+1)-dimensional VC-DJKM equation are also given.

基于Hirota双线性方法,利用拓展的同宿呼吸检验法得到了(3+1)维变系数Kudryashov-Sinelshchikov(K-S)方程的同宿呼吸波解,对该解的参数选取合适的数值,可得到不同结构的同宿呼吸波.通过对同宿呼吸波解的周期取极限,推导出方程的怪波解.最后,构造出一个特殊的高阶多项式作为测试函数,求得该方程的一阶怪波解和二阶怪波解.

Purpose The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets. Design/methodology/approach The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method. Findings The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures. Research limitations/implications This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations. Originality/value The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

In this paper we contribute to a class of degenerate grazing-sliding bifurcations in planar Filippov systems, assuming that the unperturbed system has a generalized homoclinic loop to a fold-fold point. Depending on the type of this fold-fold point, we exhibit some codimension-2 bifurcation diagrams as well as the asymptotics of all bifurcation curves. Our work not only reports several complete unfoldings in the investigation of degenerate grazing-sliding bifurcations, but also answers the open problem about the maximum number of limit cycles bifurcating from the generalized homoclinic loop for the case of fold-fold presented in Liang et al. (2013) [30] .

In this study, we construct multi wave solutions for the (2 + 1)-dimensional Sakovich equation by utilizing the logarithmic transformation of the dependent variables and symbolic computation with the ansatz function technique. We used three different method including multi waves method, double exponential form and homoclinic breather approach. 3D graphics of the solutions in different structures are drawn for some values for the parameters. Furthermore, the Lie symmetry analysis is performed for the (2 + 1)-dimensional Sakovich equation. Using the Lie symmetry groups approach, we construct the transformation groups and vector fields. We obtain the symmetry reductions and invariant solutions of the equation via these vector fields.

In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five.

The Josephson equation ϕ ˙ = y , y ˙ = − sin ⁡ ϕ + ϵ ( a − ( 1 + γ cos ⁡ ϕ ) y ) was researched by Sanders and Cushman (1986) [12] for its phase portraits when ϵ > 0 is small by applying the averaging method. The parameter ϵ can actually be large or even any real number in the practical application of this model. When | ϵ | is not small, we cannot apply the averaging method because the system is not near-Hamiltonian. For general ϵ ∈ R , we present complete dynamics and more complex bifurcations of the Josephson equation in T S 1 , including saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic loop bifurcation, two-saddle heteroclinic loop bifurcation, upper saddle connection bifurcation and lower saddle connection bifurcation. Moreover, we prove the monotonicity of bifurcation functions with respect to parameters and the nonexistence of a two-saddle heteroclinic loop for all a ≠ 0 .

We study the dynamics of a piecewise-smooth system of differential equations for which the existence of a strange Lorenz-type attractor had been rigorously proved previously and bifurcation mechanisms of its birth had been obtained. In this work we discuss the destruction of this attractor due to the appearance of sliding motions in its structure. Using qualitative and numerical methods, we study a complex sequence of attractor bifurcations that leaves in the system a globally stable limit cycle. We show that this sequence is based on C-bifurcations and bifurcations of multi-loop homoclinic trajectories.

The homoclinic bifurcations and nonplanar chaotic waves in axially moving beam (AMB) under thermal excitation are investigated. By the multiple scale technique, the equivalent nonlinear system is derived to explore qualitatively the dynamical characteristics of AMB system for the case of primary resonance. Using Melnikov approach as well as geometric analysis, the criterion for homoclinic chaos and complex nonplanar motions for AMB system is discussed. The theoretical predictions are tested by the numerical approach. For the design and application of the AMB, some inspiration and guidance are provided by the results from theory and simulation.

In this paper, based on a bilinear differential equation, we study the breather wave solutions by employing the extended homoclinic test method. By constructing the different forms, we also consider the interaction solutions. Furthermore, it is natural to analyse dynamic behaviors of three-dimensional plots.

The well-known Poincare-Bendixson theorem tells us that the structure of the omega-limit sets of planar autonomous differential equations can be described by fixed points, limit cycles, or finite number of fixed points together with homoclinic and heteroclinic orbits connecting them. However, this is very different for planar nonautonomous differential equations. In this paper, we study the omega-limit sets of three classes of planar nonautonomous differential equations, and their corresponding dynamical behavior. (i) The first type is linear, the omega-limit set is an annulus, the Lyapunov exponents are zero, there is no periodic orbit except for one fixed point, and the orbits are transitive in the omega-limit sets, implying the existence of infinitely many transitive components depending on initial conditions. (ii) The second type has butterfly-like omega-limit sets, two zero Lyapunov exponents, and neither periodic orbits nor sensitive dependence on initial conditions. (iii) The last type has a unique omega-limit set (a global attractor) independent on initial conditions, and has two negative Lyapunov exponents, where the omega-limit set is an annulus or a subset (homeomorphic to a disk) of an annulus for different parameter regions.

We prove the existence of weak solutions for an anisotropic homoclinic discrete nonlinear system. Suitable Hilbert spaces and norms are constructed. The proof of the main result is based on a minimization method. We also extend the problem by using generalized penality and source functions.

In this paper, we investigate multiple existence of homoclinic solutions for a periodically perturbed N-dimensional autonomous differential equation with a degenerate homoclinic solution of degeneracy degree d. Known results were obtained with a functional perturbation, which is regarded as an infinite-dimensional parameter. In this paper we consider a single parameter perturbation, a special form of former's functional perturbation, and prove that the single parameter is enough to unfold all possibilities of linearly independent homoclinic solutions bifurcated from the unperturbed degenerate homoclinic one, which actually improves the known results. Furthermore, we prove that those homoclinic solutions are all transversal, showing co-existence of multiple chaotic motions.

The multipulse homoclinic orbits and chaotic dynamics of a reinforced composite plate with the carbon nanotubes (CNTs) under combined in-plane and transverse excitations are studied in the case of 1 : 1 internal resonance. The method of multiple scales is adopted to derive the averaged equations. From the averaged equations, the normal form theory is applied to reduce the equations to a simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method proposed by Haller and Wiggins is utilized to examine the global bifurcations and chaotic dynamics of the CNT-reinforced composite plate. The analytical results demonstrate that the multipulse Shilnikov-type homoclinic orbits and chaotic motions exist in the system. Homoclinic trees are constructed to illustrate the repeated bifurcations of multipulse solutions. In order to verify the theoretical results, numerical simulations are given to show the multipulse Shilnikov-type chaotic motions in the CNT-reinforced composite plate. The results obtained here imply that the motion is chaotic in the sense of the Smale horseshoes for the CNT-reinforced composite plate.

For a planar analytic near-Hamiltonian system, whose unperturbed system has a family of periodic orbits filling a period annulus with the inner boundary an elementary centre and the outer boundary a homoclinic loop through a nilpotent singularity of arbitrary order, we characterize the coefficients of the terms with degree greater than or equal to 2 in the expansion of the first order Melnikov function near the homoclinic loop. Based on these expression of the coefficients, we discuss the limit cycle bifurcations and obtain more number of limit cycles which bifurcate from the family of periodic orbits near the homoclinic loop and the centre. Finally, as an application of our main results we study limit cycle bifurcation of a (m + 1)th order Lienard system with an elliptic Hamiltonian function of degree 4, and improve the lower bound of the maximal number of the isolated zeros of the related Abelian integral for any m ≥ 4.

or the second-order differential equation ẍ + f(t) ẋ + g(t)x = 0, the method of Lyapunov functions is used to obtain sufficient conditions for the existence of homoclinic trajectories, i.e., solutions x(t), ẋ (t) satisfying the conditions limt→±∞x(t) = 0 and limt→±∞ẋ (t) = 0. The specific case in which all the solutions of this differential equation are homoclinic is considered.

Under investigation in this letter is an extended (3 + 1)-dimensional Jimbo–Miwa (eJM) equation, which can be used to describe many nonlinear phenomena in mathematical physics. With the aid of Hirota bilinear method and long-wave limit method, M-order lumps which describe multiple collisions of lumps are derived. The propagation orbit, velocity and extremum of the 1-order lump solutions on (x, y) plane are investigated in detail. Resorting to the extended homoclinic test technique, we obtain the breather–kink solutions, rational breather solutions and rogue wave solutions for the eJM equation. Meanwhile, through analysis and calculation, the amplitude and period of breather–kink solutions increase with p increasing and the extremum of rational breather solution and rogue waves are also derived. T-order breathers are obtained by means of choosing appropriate complex conjugate parameters on N-soliton solutions. Periods of the 1-order breather solutions on the (x, y) plane are determined by $$k_{12}$$ and $$k_{12}p_{11}+k_{11}p_{12}$$, while locations are determined by $$k_{11}$$ and $$k_{11}p_{11}-k_{12}p_{12}$$. Furthermore, hybrid solutions composed of the kink solitons, breathers and lumps for the eJM equation are worked out. Some figures are given to display the dynamical characteristics of these solutions.

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.

The multiplicity of homoclinic solutions is obtained for a class of thep-Laplacian Hamiltonian systems$\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))-a(t)|u(t)|^{p-2}u(t)+ \nabla W(t,u(t))=0$ddt(|u˙(t)|p−2u˙(t))−a(t)|u(t)|p−2u(t)+∇W(t,u(t))=0via variational methods, where$a(t)$a(t)is neither coercive nor bounded necessarily and$W(t,u)$W(t,u)is under new concave–convex conditions. Recent results in the literature are generalized even for$p=2$p=2.

In this paper, Melnikov analysis of chaos in a simple SIR model with periodically or stochastically modulated nonlinear incidence rate and the effect of periodic and bounded noise on the chaotic motion of SIR model possessing homoclinic orbits are detailed investigated. Based on homoclinic bifurcation, necessary conditions for possible chaotic motion as well as sufficient condition are derived by the random Melnikov theorem, and to establish the threshold of bounded noise amplitude for the onset of chaos.

The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What's more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor.

In this work, some new effective methods for suppressing homoclinic chaos in a weak periodically excited non-smooth oscillator are studied, and the main idea is to modify slightly the Melnikov function such that the zeros are eliminated. Firstly, a general form of planar piecewise-smooth oscillators is given to approximatively model many nonlinear restoring force of smooth oscillators subjected to all kinds of damping and periodic excitations. In the absence of controls, the Melnikov method for non-smooth homoclinic trajectories within the framework of a piecewise-smooth oscillator is briefly introduced without detailed derivation. This analytical tool is useful to detect the threshold of parameters for the existence of homoclinic chaos in the non-smooth oscillator. After some methods of state feedback control, self-adaptive control and parametric excitations control are, respectively, considered, sufficient criteria for suppressing homoclinic chaos are derived by employing the Melnikov function of non-smooth systems. Finally, the effectiveness of strategies for suppressing homoclinic chaos is analytically and numerically demonstrated through a specific example.

We revisit the analysis by R.A. Gardner of convergence of spectra of periodic traveling waves in the homoclinic, or infinite-period limit, extending his results to the case of essential rather than point spectra of the limiting homoclinic wave. Notably, convergence to essential spectra is seen to be of algebraic rate with respect to period as compared to the exponential rate of convergence to point spectra. In the course of the analysis, we show not only convergence of spectrum but also convergence of an appropriate renormalization of the associated periodic Evans function to the Evans function for the limiting homoclinic wave, a fact that is useful for numerical investigations.

In this paper, we systematical study the rich dynamics and complex bifurcations of a non-monotonic predator–prey system with a constant releasing rate for the predator. We prove that the system can have at most three positive equilibria, and can undergo a sequence of bifurcations, including transcritical, saddle-node, Hopf, degenerate Hopf, double limit cycle, saddle-node homoclinic bifurcation (or homoclinic loop with a saddle-node), cusp bifurcation of codimension 2, and Bogdanov–Takens bifurcation of codimension 2 and 3. And the system can generate very rich dynamics, such as the existence of a semi-stable limit cycle, multiple coexistent periodic orbits, homoclinic loops, etc. Moreover, our results show that the dynamical behaviors highly rely on the constant releasing rate of predators and the initial conditions. That is, there exists a critical value of the constant releasing rate of predators such that (i) when the constant releasing rate is greater than the critical value, the prey goes to extinction for all admissible initial populations of both species; (ii) when the constant releasing rate is less than the critical value, the prey can always coexist with the predator. Numerical simulations are presented to verify the main results.

Global orbits connect the saddle points in an infinite period through the homoclinic and heteroclinic types of manifolds. Different from the periodic movement analysis, it requires special strategies to obtain expression of the orbit and detect the associated profound dynamic behaviors, such as chaos. In this paper, a global dynamic frequency method is applied to detect the homoclinic and heteroclinic bifurcation of the complicated nonlinear systems. The so-called dynamic frequency refers to the newly introduced frequency that varies with time t, unlike the usual static variable. This new method obtains the critical bifurcation value as well as the analytic expression of the orbit by using a standard five-step hyperbolic function-balancing procedure, which represents the influence of the higher harmonic terms on the global orbit and leads to a significant reduction of calculation workload. Moreover, a new homoclinic manifold analysis maps the periodic excitation onto the target global manifold that transfers the chaos discussion of non-autonomous systems into the orbit computation of the general autonomous system. That strategy unifies the global bifurcation analysis into a standard orbit approximation procedure. The numerical simulation results are shown to compare with the predictions.

The present study investigates the lump, one-stripe, lump-stripe, and breather wave solutions to the (2+1)-dimensional Sawada-Kotera equation using the Hirota bilinear method. For lump and lump-stripe solutions, a quadratic polynomial function, and a quadratic polynomial function in conjunction with an exponential term are assumed for the unknown function f giving the solution to the mentioned equation, respectively. On the other hand, only an exponential function is considered for one-stripe solutions. Besides, the homoclinic test approach is adopted for constructing breather wave solutions. The propagations of the attained lump, lump-stripe, and breather wave solutions are shown through some graphical illustrations. The graphical outputs demonstrate that the lump wave moves along the straight line y=−x and exponentially decreases away from the origin of the spatial domain. On the other hand, lump-kink solutions illustrate the fission and fusion interaction behaviors upon the selection of the free parameters. Fission and fusion processes show that the stripe soliton splits into a stripe soliton and a lump soliton, and then the lump soliton merges into one stripe soliton. In addition, the achieved breather waves illustrate the periodic behaviors in the xy-plane. The outcomes of the study can be useful for a better understanding of the physical nature of long waves in shallow water under gravity., (2+1)-dimensional Sawada-Kotera equation; Hirota bilinear method; Lump solutions; Lump-stripe solutions; Homoclinic test approach; Breather wave solutions

The paper deals with problem of ordering coexistence of homoclinic trajectories for maps of segment into itself, which are generated by a certain continuous function of the segment.