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This paper is concerned with existence, uniqueness and asymptotic behavior of traveling wave fronts for a generalized Fisher equation with nonlocal delay. The existence of traveling wave fronts is established by linear chain trick and geometric singular perturbation theory. The strategy is to reformulate the problem as the existence of a heteroclinic connection in R 4 . The problem is then tackled by using Fenichel’s invariant manifold theory. The asymptotic behavior and uniqueness of traveling wave fronts are also obtained by using standard asymptotic theory and sliding method.

In this paper we prove that the Camassa–Holm equation with quartic nonlinearity is non-integrable via the Painleve method. The orbital stability of solitary waves for this equation is investigated by constructing a functional extremum problem. This result demonstrates that the resulting solitary wave is unstable when its speed lies in the narrow region of the critical value that connects with the bifurcation condition. In contrast when the speed surpasses the narrow region, the solitary wave is stable. In addition, the stable solitary wave turns into a chaotic state when is driven externally. If a damping term controller is added to the perturbed equation, the solitary wave can also propagate stably under a certain condition. Finally our numerical results show that the perturbed equation is not well controlled when a certain resonant-frequency occurs and is well controlled with a smaller wave speed as well as a higher nonlinear convection.

The traveling wave solutions to the Dullin–Gottwald–Holm equation (called DGH equation) are classified by an improved qualitative analysis method. Meanwhile, the influence of the parameters on the traveling wave forms is specifically considered. The equation is shown to admit more traveling wave forms solutions, especially new solutions such as stumpons and fractal-like waves are first given. We also point out that the smooth solutions can converge to non-smooth ones under certain conditions. Furthermore, the new explicit forms of peakons with period are obtained.

In this paper, the qualitative analysis methods of a dynamical system are used to investigate the peaked wave solutions of K(2, 2) equation with osmosis dispersion. The phase portrait bifurcation of the traveling wave system corresponding to the equation is given. The explicit expressions of the peaked solitary wave solution and the periodic cusp wave solution are obtained by using the portraits. The graph of the solution is given with the numerical simulation.

This paper is involved with control and synchronization of a new four-dimensional energy resources system with unknown parameters. Based on the Lyapunov stability theorem, by designed adaptive controllers and parameter update laws, this system is stabilized to unstable equilibrium and synchronizations between two systems with different unknown system parameters are realized Numerical simulations are given for the purpose of illustration and verification.

In this paper, using the idea of the successive dislocation feedback method, a new time-delayed feedback control method called the successive dislocation time-delayed feedback control (SDTDFC) is designed. Firstly, the idea of SDTDFC is introduced. Then some analytic sufficient conditions of the chaos control from the SDTDFC approach are derived for stabilization. Finally, some established results are further clarified via a case study of the Lorenz system with the numerical simulations.

In this paper, we study an m-part uniform Cantor set with its rational translation. We give the fractal structure and the formula of the box-counting dimension of the intersection I(t). We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from translating the length t with its n-base expansion.

In this paper, the generalized Ostrovsky equation is introduced. Using a direct and effective method, some new solitary solutions to the generalized Ostrovsky equation, such as compacton solutions, multi-compacton solutions and compact-like kink solutions can be obtained. The homogenous balance (HB) method is used to obtain the Backlund transformation. And some new solitary solutions, particularly new double symmetric peakon solutions, are given by the transformation.

This paper discusses the problem of hyper-chaos control of an optical system. Based on Lyapunov stability theory, a non-autonomous feedback controller is designed. The proposed controller ensures that the hyper-chaotic system will be asymptotically stable. Numerical simulation of the original and the controlled system is provided to show the effectiveness of our method.

The number and distribution of limit cycles of a perturbed Z 4 -equivariant Hamiltonian system are studied in this paper. The existence theory and stability theory of singular close orbits are applied to study the given perturbed system. By using the small parametric perturbation skills of differential equations, we find that the perturbed Z 4 -equivariant system has at least 20 limit cycles. The distribution of the above 20 limit cycles is also given.

In this paper, the problem of control for the energy resource chaotic system is considered. Two different method of control, feedback control (include linear feedback control, non-autonomous feedback control) and adaptive control methods are used to suppress chaos to unstable equilibrium or unstable periodic orbits. The Routh–Hurwitz criteria and Lyapunov direct method are used to study the conditions of the asymptotic stability of the steady states of the controlled system. The designed adaptive controller is robust with respect to certain class of disturbances in the energy resource chaotic system. Numerical simulations are presented to show these results.

This paper establishes a new energy resources demand–supply system for two regions of China. The dynamics behavior of this system is also analyzed. Lyapunov exponents and homoclinic orbits of this system can be obtained. The energy resources demand–supply system is chaotic by analytically demonstration and Lyapunov exponents, which displays a two-layer attractor.

Two generalizations of the Camassa–Holm system associated with the singular analysis are proposed for Painleve integrability properties and the extensions of already known analytic solitons. A remarkable feature of the physical model is that it has peakon solution which has peak form. An alternative WTC test which allowed the identifying of such models directly if formulated in terms of inserting a formed ansatz into these models. For the two models have Painleve property, Painleve–Backlund systems can be constructed through the expansion of solitons about the singularity manifold. By the implementations of Maple, plentiful new type solitonic structures and some kink waves, which are affected by the variation of energy, are explored. If the energy is infinite in finite time, there will be a collapse in soliton systems by direct numerical simulations. Particularly, there are two collapses coexisting in our regular solitons, which occurred around its central regions. Simulation shows that in the bottom of periodic waves arises the non-zero parts of compactons and anti-compactons. We also get floating solitary waves whose amplitude is infinite. In contrary to which a finite-amplitude blow-up soliton is obtained. Periodic blow-ups are found too. Special kinks which have periodic cuspons are derived.

This paper discusses a new energy resource chaotic system. It investigates basically dynamical behaviors of this new system. It also addresses the synchronization problem of two energy resource systems in the presence of different unknown system parameters. Based on Lyapunov stability theory, an adaptive control law is derived to make the states of two energy resource systems with different unknown system parameters asymptotically synchronized. Numerical simulations are given to validate the synchronization approach.

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for the classification purpose. In this paper, we pursue this study on a class of Moran sets with their rational translates. We also get the fractal structure of intersection I ( x , y ) of a class of Moran sets with their rational translates, and the formula of the box-counting dimension. We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from shifting vector with the expansion in fraction of ( x , y ). Concretely, when ( x , y ) has a finite expansion in fraction, a very brief calculation formula of the measure is given.

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for a classification purpose. In this paper, we pursue this study on the Sierpinski carpet with its rational translate. We also get the fractal structure of intersection I ( x , y ) of the Sierpinski carpet with its translate. We find that the packing measure of these sets forms a discrete spectrum whose non-zero values come only from shifting numbers with a finite triadic expansion. Concretely, when x and y have a finite triadic expansion, a very brief calculation formula of the measure is given.

Considering a time-delay in the receiver as compared with the transmitter, we addresses a practical issue in chaos synchronization of WINDMI system which is based on the Lyapunov stabilization theory and matrix measure, such that the state of the slave system at time t is asymptotically synchronizing with the master at time t − τ. The Mathematical software is used to prove the effectiveness of this method.

The analysis qualitative methods of planar dynamical systems are used to study the peaked solitary wave solutions for Degasperis–Procesi equation with the dispersion term. By using the phase portrait bifurcation of traveling wave system, periodic wave solutions and solitary wave solutions are constructed in two different ways, and their convergence is showed when g varies. The general explicit expression of peaked solitary wave solutions is obtained under some parameter conditions.

Some special solutions of the Toda lattice model with a transversal degree of freedom are obtained. With the aid of Mathematica and Wu elimination method, more explicit solitary wave solutions, including compacton solutions, multiple compacton solutions, peakon solutions, as well as periodic solutions are found in this paper.

In this paper, we introduce the concept of nonlinear intensity and study the new type solitary wave solutions of nonlinear intensity Klein–Gordon equation. Applying the improved generalized projective Riccati method, abundant exact solitary wave solutions are obtained. By using ansatzes, nonsymmetrical compacton, multi-compacton solutions, pattern solitary wave solutions and singular periodic wave solutions are found. Then, we discuss the changes of compacton solutions under various nonlinear intensity parameters and get the compacton solutions of higher dimensions nonlinear intensity Klein–Gordon equation.

In this paper, we study a sort of chaotic system—Newton–Leipnik system which possesses two strange attractors. The static and dynamic bifurcations of the system are studied. The chaos controlling is performed by a simpler linear controller, and numerical simulation of the control is supplied. At the same time, Lyapunov exponents of the system show that the result of the chaos controlling is right.

We study a sort of nonlinear reaction diffusion equation based on the modified Korteweg–de Vries (mKdV) equation with a higher order singularly perturbing term as the Kuramoto–Sivashinsky (KS) equation, called mKdV–KS equation. Special attention is paid to the question of the existence of solitary wave solutions. Based on the analogue between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, from geometric singular perturbation point of view, we prove that solitary wave persists when the perturbation parameter is suitably small. This argument does not require an explicit expression for the original mKdV solitary wave solution.

In this paper, for the contracting similarities S 0 ( x ) = x 5 , S 1 ( x ) = x + λ 5 , S 2 ( x ) = x + 2 5 , S 3 ( x ) = x + 4 - λ 5 and S 4 ( x ) = x + 4 5 , where rational number λ ∈ [ 0 , 2 3 ] , the Hausdorff dimension and the structure of Eλ, an invariant set with respect to S0, S1, S2, S3, S4 were studied and some new results were reported. Relation to El Naschie’s quantum fractal is also discussed.

Let K(λ1, λ2), the symmetry Cantor sets, be the attractor of an iterated function system {f1, f2, f3} on the line, where f1(x) = λ1x, f 2 ( x ) = λ 2 x + 1 - λ 2 2 , f 3 ( x ) = 1 - λ 1 + λ 1 x , x ∈ [0, 1]. In this paper, we proved that if 1 - 2 λ 1 - λ 2 2 ⩾ λ , where λ ≡ max{λ1, λ2}, then the exact Hausdorff centered measure Cs of K(λ1, λ2) equals 1, where s is the Hausdorff dimension of K(λ1, λ2).

We introduce a fifth-order K ( m , n ,1) equation with nonlinear dispersion to obtain multi-compacton solutions by Adomian decomposition method. Using the homogeneous balance (HB) method, we derive a Backlund transformation of a special equation K (2,2,1) to determine some solitary solutions of the equation. To study the stability of multi-compacton solutions in K ( m , n ,1) and to obtain some conservation laws, we present a similar fifth-order equation derived from Lagrangian. We finally show the linear stability of all obtained multi-compacton solutions.

This paper introduces a new method for the coupled dynamos dynamical system, which can be applied to the decision of the chaotic behavior of the system. And research the tracing control of the chaos for the coupled dynamos dynamical system by gradually changing the driving parameter for the chaos. With the different design of controllers, the numerical simulation results show the relation between the chaotic behavior and the changes of the parameter value. Furthermore, the result shows the difference of the controllers. In the mean time, it reveals the process of the orbit’s gradual changing with the parameter value.

In this paper, we consider generalized Camassa–Holm equations and the generalized weakly dissipative Camassa–Holm equations and derive some new exact peaked solitary wave solutions. For m=3, where m is representative of the strength of the nonlinearity, we give two types new exact traveling wave solutions of the generalized weakly dissipative Camassa–Holm equations.