A new method, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solution to nonlinear evolution equation (NEE) is proposed. (1+1)-dimensional Boussinesq equation is used as an example to illustrate the effectiveness of the suggested method. Rational homoclinic wave solution, a new family of two-wave solution, is obtained by inclined periodic homoclinic breather wave solution and is just a rogue wave solution. This result shows that rogue wave originates by the extreme behaviour of homoclinic breather wave in (1+1)-dimensional nonlinear wave fields.

A lump-type solution of the (2 + 1)-dimensional generalised fifth-order Korteweg–de Vries (KdV) equation is obtained from the two-soliton solution by applying the parametric limit method. Some theorems and corollaries about the superposition behaviour between lump solutions and different forms of $N$-soliton ($N \rightarrow\infty$) solutions are constructed, and detailed proofs are given. Besides,we give a large number of examples and spatial evolution graphics to illustrate the effectiveness of the described theorems and corollaries. Some new nonlinear phenomena and superposition behaviour, such as rational-exponential type, rational-cosh-cos type, rational-sin type, rational-logarithmic type etc., are simulated and shown for the first time. Finally, we also illustrate the superposition between high-order lump-type solutions and $N$-soliton solutions.

By studying the time series of logistic maps, dark lines in bifurcation diagrams and cobweb diagrams, characteristics of sequential iterations of the map are found. Before merging together two chaotic bands, sequential iterations of the map present an ordered state. After that, with the instability of the hyperstable periodic orbit, sequential iterations of the map appear disordered. Therefore, the statistical results of time correlation of direction phase are introduced to describe the transition of the collective behaviour. Based on the two-dimensional coupled map lattice, the relationship between bifurcation parameters and order parameters with the change of coupling strength is studied. We show that when the coupling strength is weak, the critical bifurcation parameters are positively correlated with the coupling strength. When the coupling strength is large, the phase order of the system is not affected by the coupling strength. The transition of collective behaviour in a modular network is also studied. By fixing modularity and bifurcation parameters, with the change of coupling strength, the collective behaviour presents a transition process from spatiotemporal chaos to phase-ordered state. There are periodic orbits in the spatiotemporal chaotic region. A phase synchronisation region can be present in the antiphase synchronisation region. Furthermore, there exist multistable solutions in the region.