Transitions are to be found throughout the natural world. The laws of nature are usually
represented by differential equations, which can be regarded as dynamical systems—both …

The main objectives of this article are to introduce stochastic parameterizing manifolds and
to study the dynamical transitions of the two‐dimensional stochastic Swift‐Hohenberg …

This paper deals with the classification of transition phenomena in the most basic dissipative
system possible, namely, the 1D reaction–diffusion equation. The emphasis is on the …

We study the dynamic transition of the Swift-Hohenberg equation (SHE) when linear
multiplicative noise acting on a finite set of modes of the dominant linear flow is introduced …

In this paper, we study the dynamical bifurcation and final patterns of a modified Swift-
Hohenberg equation (MSHE). We prove that the MSHE bifurcates from the trivial solution to …

In this paper, we study bifurcation of the damped Kuramoto-Sivashinsky equation on an odd
periodic interval of period 2λ. We fix the control parameter α∈(0, 1) and study how the …

In this paper, we study the primary instability of the damped Kuramoto–Sivashinsky equation
under a periodic boundary condition. We prove that it bifurcates from the trivial solution to an …

The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper,
we study the dynamic transitions of the Swift-Hohenberg equation with a third-order …

In this paper, we study the dynamical bifurcation of the modified Swift-Hohenberg equation
on a periodic interval as the system control parameter crosses through a critical number …

In this paper, we prove that the generalized Swift–Hohenberg equation bifurcates from the
trivial states to an attractor as the control parameter α passes through critical points. The …