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1. Global Bifurcation for Corotating and Counter-Rotating Vortex Pairs

   https://doi.org/10.1007/s00220-023-04741-6  

   García, Claudia; Haziot, Susanna V. (June 2023, Communications in Mathematical Physics)

   Abstract The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu (in Commun Math Phys 350(2):699–747, 2017) via a desingularization of a pair of point vortices. In this paper, we construct a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of a trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland to allow for the singularity at the bifurcation point. For both the corotating and the counter-rotating pairs, along the global curve of solutions either the angular fluid velocity vanishes or the two patches self-intersect. 

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2. Analytical construction of soliton families in one‐ and two‐dimensional nonlinear Schrödinger equations with nonparity‐time‐symmetric complex potentials

   https://doi.org/10.1111/sapm.12383  

   Yang, Jianke (April 2021, Studies in Applied Mathematics)

   Abstract The existence of soliton families in nonparity‐time‐symmetric complex potentials remains poorly understood, especially in two spatial dimensions. In this article, we analytically investigate the bifurcation of soliton families from linear modes in one‐ and two‐dimensional nonlinear Schrödinger equations with localized Wadati‐type nonparity‐time‐symmetric complex potentials. By utilizing the conservation law of the underlying non‐Hamiltonian wave system, we convert the complex soliton equation into a new real system. For this new real system, we perturbatively construct a continuous family of low‐amplitude solitons bifurcating from a linear eigenmode to all orders of the small soliton amplitude. Hence, the emergence of soliton families in these nonparity‐time‐symmetric complex potentials is analytically explained. We also compare these analytically constructed soliton solutions with high‐accuracy numerical solutions in both one and two dimensions, and the asymptotic accuracy of these perturbation solutions is confirmed. 

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3. A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

   https://doi.org/10.1063/5.0208538  

   Anurag, Atul; Goodman, Roy H; O'Grady, Ellison K (June 2024, Physics of Fluids)

   We introduce a new reduction of the motion of three point vortices in a two-dimensional ideal fluid. This proceeds in two stages: a change of variables to Jacobi coordinates and then a Nambu reduction. The new coordinates demonstrate that the dynamics evolve on a two-dimensional manifold whose topology depends on the sign of a parameter κ2 that arises in the reduction. For κ2>0, the phase space is spherical, while for κ2<0, the dynamics are confined to the upper sheet of a two-sheeted hyperboloid. We contrast this reduction with earlier reduced systems derived by Gröbli, Aref, and others in which the dynamics are determined from the pairwise distances between the vortices. The new coordinate system overcomes two related shortcomings of Gröbli's reduction that have made understanding the dynamics difficult: their lack of a standard phase plane and their singularity at all configurations in which the vortices are collinear. We apply this to two canonical problems. We first discuss the dynamics of three identical vortices and then consider the scattering of a propagating dipole by a stationary vortex. We show that the points dividing direct and exchange scattering solutions correspond to the locations of the invariant manifolds of equilibria of the reduced equations and relate changes in the scattering diagram as the circulation of one vortex is varied to bifurcations of these equilibria. 

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4. A global bifurcation organizing rhythmic activity in a coupled network

   https://doi.org/10.1063/5.0089946  

   Medvedev, Georgi S.; Mizuhara, Matthew S.; Phillips, Andrew (August 2022, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   We study a system of coupled phase oscillators near a saddle-node on invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using Ott–Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation from noncontractibile ones after bifurcation. Both families are stable for the model at hand. 

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5. Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability

   https://doi.org/10.1016/j.physd.2024.134058  

   Chekroun, Mickaël D; Liu, Honghu (April 2024, Physica D: Nonlinear Phenomena)

   Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics. To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems. These dynamical insights enable us in turn to design a stochastic model whose solutions---as the delay parameter drifts slowly through its critical values---produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping ‘‘points’’ beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena. 

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