Related papers: A Formal Geometric Blow-up Method for Pattern Form…
We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to…
We outline a general theory for the analysis of flow-distributed standing and travelling wave patterns in one-dimensional, open plug-flows of oscillatory chemical media. We treat both the amplitude and phase dynamics of small and…
We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an…
The article is devoted to the results of a phase topology research on a generalized mathematical model, which covers such two problems as dynamics of two point vortices enclosed in a harmonic trap in a Bose-Einstein condensate and dynamics…
Bifurcations of periodic orbits as an external parameter is varied are a characteristic feature of generic Hamiltonian systems. Meyer's classification of normal forms provides a powerful tool to understand the structure of phase space…
The space of smooth rational cubic curves in projective space $\PP^r$ ($r\ge 3$) is a smooth quasi-projective variety, which gives us an open subset of the corresponding Hilbert scheme, the moduli space of stable maps, or the moduli space…
Inspired by an example of Grebogi et al [1], we study a class of model systems which exhibit the full two-step scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold [2]. The specific structure of these models allows a…
This paper is devoted to the study of the dynamical behavior of the critically dissipative quasi-geostrophic equation in $\textbf{T}^2$. We prove that this system possesses time-dependent periodic solutions, bifurcating from a smooth steady…
Model studies indicate that many climate subsystems, especially ecosystems, may be vulnerable to 'tipping': a 'catastrophic process' in which a system, driven by gradually changing external factors, abruptly transitions (or 'collapses')…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
We study the bifurcation of planar patterns and fast-moving pattern interfaces in an asymptotic long-wave model for the three-dimensional B\'enard-Marangoni problem, which is close to a Turing instability. We derive the model from the full…
We consider the finite-time blow-up dynamics of solutions to the self-dual Chern-Simons-Schr\"odinger (CSS) equation (also referred to as the Jackiw-Pi model) near the radial soliton $Q$ with the least $L^{2}$-norm (ground state). While a…
We study, both theoretically and experimentally, the dynamical response of Turing patterns to a spatio-temporal forcing in the form of a travelling wave modulation of a control parameter. We show that from strictly spatial resonance, it is…
To obtain explicit understanding of the behavior of dynamical systems, geometrical methods and slow-fast analysis have proved to be highly useful. Such methods are standard for smooth dynamical systems, and increasingly used for continuous,…
Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties in real-world systems, especially when the solution is not unique or exhibits sudden qualitative changes as parameters vary.…
In this paper, the dynamic of inertial capsules into microfluidic bifurcations is studied. The fluid evolution is based on the solution of the BGK -- lattice Boltzmann scheme including a forcing term accounting for immersed geometries. The…
Blowing up a point p in a manifold M builds a new manifold M' in which p is replaced by the projectivization of the tangent space of M at p. This well-known operation also applies to fixed points of diffeomorphisms, yielding continuous…
We study linear stability of exponential periodic solutions of a system of singular amplitude equations associated with convective Turing bifurcation in the presence of conservation laws, as arises in modern biomorphology models, binary…
The well known concept, to reduce the spatio-temporal dynamics beyond instabilities of trivial states to amplitude modulated patterns, is reviewed from the point of view of a formal perturbation expansion for general dissipative partial…
In this work we employ a recently proposed bifurcation analysis technique, the deflated continuation algorithm, to compute steady-state solitary waveforms in a one-component, two dimensional nonlinear Schr\"odinger equation with a parabolic…