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This paper deals with a class of one element n-degree polynomial differential equations. By the fixed point theory, we obtain n periodic solutions of the equation. This paper generalizes some related conclusions of some papers.

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system, studies some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponents, bifurcation diagram and the hyperchaotic attractor evolving into periodic, quasi-periodic dynamical behaviours by varying parameter κ. Furthermore, effective linear feedback control method is used to suppress hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results.

A new idea is put forward to modify the Clarkson–Kruskal (CK) direct method. Using the usual CK direct method to a coupled KdV system, two types of usual similarity reductions can be obtained. However, the application of the modified CK direct method leads to three types of new similarity reductions different from the usual ones.

Using the tanh method and a variable separated ordinary difference equation method to solve the double sine-Gordon equation, we derive some new exact travelling wave solutions, especially a new type of noncontinuous solitary wave solutions. These noncontinuous solitary wave solutions are verified by using the conservation law theory.

In this paper, Lie symmetry is investigated for a new integrable coupled Korteweg–de Vries (KdV) equation system. Using some symmetry subalgebra of the equation system, we obtain five types of the significant similarity reductions. Abundant solutions of the coupled KdV equation system, such as the solitary wave solution, exponential solution, rational solution and polynomial solution, etc. are obtained from the reduced equations. Especially, one type of group-invariant solution of reduced equations can be acquired by means of the Painleve I transcendent function.

We investigate the formations of wurtzite (WZ) SiC nano polytypes in zinc blende (ZB) SiC nanofilms hetero-grown on Si-(100) substrates via low pressure chemical vapor deposition (LPCVD) by adjusting the Si/C ratio of the introduced precursors. Through SEM, TEM, and Raman characterizations, we find that the nanofilms consist of discrete WZ SiC nano polytypes and ZB SiC polytypes composed of WZ polytypes (WZ + ZB) and disordered ZB SiC polytypes, respectively, according to Si/C ratios of 0.5, 1.5, and 3. We attribute the WZ polytype formation to being due to a kinetic mechanism based on the Si/C surface saturation control.

Under different system parameters, some new integrable models of the generalized modified Dullin-Gottwald-Holm are obtained by Painlev analysis. Using the auto-Backlund transformation, solitary wave solutions of these integrable models are obtained.

This paper introduces a new three dimensional autonomous system with five equilibrium points. It demonstrates complex chaotic behaviours within a wide range of parameters, which are described by phase portraits, Lyapunov exponents, frequency spectrum, etc. Analysis of the bifurcation and Poincare map is used to reveal mechanisms of generating these complicated phenomena. The corresponding electronic circuits are designed, exhibiting experimental chaotic attractors in accord with numerical simulations. Since frequency spectrum analysis shows a broad frequency bandwidth, this system has perspective of potential applications in such engineering fields as secure communication.

This paper first investigates the projective synchronisation problem with non-delayed and delayed coupling between drive-response dynamical networks consisting of identical nodes and different nodes. Based on Lyapunov stability theory, several nonlinear controllers are applied to achieve the projective synchronisation between the drive-response dynamical networks; simultaneously the topological structure of the drive dynamical complex networks can be exactly identified. Moreover, numerical examples are presented to verify the feasibility and effectiveness of the theorems.

The new autonomous system with only three equilibrium points is introduced. This system does not belong to the generalized Lorenz systems. The novel attractors are observed over a large range of parameters, which have rarely been reported in previous work. As an important component in chaotic signal generators, a physical circuit has been designed. The experimental results are in agreement with numerical simulations. More significantly, spectral analysis shows that the system has an extremely broad frequency spectral bandwidth in 0–131.6 Hz, without investigating any possible electronic techniques, which is more desirable for secure communications.

New exotic solitary wave and the painlevé integrability of one type of the nonlinear dispersive generalized DGH equation are studied. By Painlevé analysis, we discover that the nonlinear dispersive generalized DGH equation with m＝2 is integrable, which is a new integrable equation. By the new variable transformation and the auto-Backlund transformation，we obtain abundant exotic solitary wave solutions, such as compactons, peakons, new double solitary waves with peak points, and double solitary waves with blow-up points.

Generalized projective synchronization (GPS) between two complex networks with time-varying coupling delay is investigated. Based on the Lyapunov stability theory, a nonlinear controller and adaptive updated laws are designed. Feasibility of the proposed scheme is proven in theory. Moreover, two numerical examples are presented, using the energy resource system and Lu's system [Physica A 382 (2007) 672] as the nodes of the networks. GPS between two energy resource complex networks with time-varying coupling delay is achieved. This study can widen the application range of the generalized synchronization methods and will be instructive for the demand–supply of energy resource in some regions of China.

In this paper we introduce one type of generalized dispersive Camassa-Holm model and make its singularity analysis. We prove that the model is Painlevé integrable by an alternative WTC-Kruskal test and obtain the Painlevé-Bcklund systems and the Bcklund transformation. Many new types of regular soliton, singular soliton, kink soliton, compacton and anti-compacton are explored. Particularly, we have found singular structures of periodic cuspon waves in kink solitons, which occur in their central regions. Based on the regular solitonic system, we do Bcklund transformation and obtain three sorts of singular solitons, namely the periodic blow-up wave with hump structure, kink soliton for the blow-up wave structure and the compacton.

This paper proposes a method of non-autonomous feedback control, and proves its validity by the stability theory. This method is used in numerical emulation of energy resource chaotic system and Arneodo system. The result indicates that it is correct and feasible.

By constructing two new Riccati equations and using the generalized Riccati method, we simplified the form and enriched the general results.Using Mathematica software, exact solutions of the variable coefficient combined KdV equation with forced term are obtained, including many kinds of solitary-wave-like solutions,guasi-periodical solutions and solitary wave solutions with variable speed.

Study one type of nonlinear equations,namely generalized Camassa-Holm equation C(n):u\-t+ku\-x+β\-1uxxt+β\-2(un+1)x+β\-3u\-x(u\+n)xx+β\- 4u(u\+n)xxx=0.Obtain abundant exact solutions by four ansasz,particularly when k≠0,we obt ain compacton solutions;while whenk=0,we obtain floating compacton solution s.At last,we also study other forms of fully nonlinear generalized Camassa-Holm equation,and their compacton solutions are governed by nonlinear equations.