Recent advances in bifurcation analysis: Theory, methods, applications and beyond…

The description of the phase space of a dynamical system has attracted the attention of the scientific community for decades. With many examples coming from the real world, the motivation to understand the underlying nature of a given dynamical system has led to a rich cooperation between the theoretical and the applied aspects of the subject. Such dynamical models may come in various sizes and shapes: they can be finite dimensional (given by the flow of a vector field or iterations of a map) or infinite dimensional (defined by the evolution operator of a PDE). Of special interest are systems whose solutions undergo topological changes upon variations on their parameters. These events are known as bifurcations and are ubiquitous in every nonlinear system that depends on parameters. These phenomena are characterized by the re-arrangement of invariant objects - such as equilibria, periodic solutions and invariant manifolds - when one or more control parameters are perturbed beyond a critical threshold. Typically, a bifurcation triggers crucial transitions from one kind of qualitative dynamics to completely new different behaviors. This may result in dramatic changes of the dynamics, including passages to chaotic regimes, transforming or creating basins of attraction, shaping and giving rise to families of multiple solution types with a particular set of spatio-temporal features and, ultimately, reorganizing the overall structure of the phase space.

While local bifurcations are well explained by means of linear analysis, normal forms and desingularization techniques, global phenomena remain a challenging topic in both continuous and discrete systems. Common tools on the subject range from reductions to Poincaré return maps in suitable cross sections to software packages to detect and path-follow the associated bifurcation sets in parameters, and even sophisticated techniques to compute the relevant invariant manifolds involved. Today, the scope of bifurcation theory has broadened to make an impact on rapidly growing branches of dynamics such as slow-fast systems, piece-wise models, delay differential equations, Hamiltonian systems, stochastic systems, as well as across the pattern formation theory. Recent discoveries in these areas have seen the emergence of new exciting types of bifurcations, some of which have yet to be addressed in all their complexity.

This is of special interest to understand the nature of systems near bifurcations in many applications, such as in laser dynamics, nerve impulses in neurons, electrochemical reactions, extinction/survival/synchronising thresholds in population models in ecology and developmental biology, fluid mechanics and celestial mechanics, to name a few examples.

The aim of this Research Topic is to present recent progresses in the study and computation of bifurcations in both theoretical and applied contexts, in order to obtain deeper insight into old and new challenges arising from these phenomena.

Both original research and review articles are expected for this issue. Some of the topics addressed in this collection include the following themes:
• Bifurcation theory of ODEs and maps.
• New bifurcation phenomena in systems with different time scales, piece-wise systems, delay differential equations, dynamical networks, and stochastic dynamical systems.
• Numerical bifurcation analysis.
• Global bifurcations and invariant manifolds.
• Reaction-diffusion and spatially extended systems.
• Homoclinic and heteroclinic phenomena.
• Bifurcations and chaos.
• Applications in physics, biology, chemistry, medicine, engineering, and social sciences.