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Periodic orbits are among the simplest non-equilibrium solutions to dynamical systems, and they play a significant role in our modern understanding of the rich structures observed in many systems. For example, it is known that embedded…
Flow instability and turbulent transition can be well explained using a new proposed theory--Energy gradient theory [1]. In this theory, the stability of a flow depends on the relative magnitude of energy gradient in streamwise direction…
Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a…
In this paper, we study the two-dimensional steady compactly supported incompressible Euler equations with free boundaries. We consider flows with constant vorticity that are perturbations of annular equilibria, in contrast to the laminar…
We study stationary stable processes related to periodic and cyclic flows in the sense of Rosinski [Ann. Probab. 23 (1995) 1163-1187]. These processes are not ergodic. We provide their canonical representations, consider examples and show…
In this work we prove that the shallow water flow, subject to strong wind forcing and linearized around an adequate stationary profile, develops for large times closed trajectories due to the propagation of Rossby waves, while Poincar\'e…
Depending on the involved physiobiological parameters, stable or unstable behavior in active fluids is observed. In this paper a rigorous analytical justification of (in-)stability within the corresponding regimes is given. In particular,…
This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In three previous papers, we introduce the notion of formal manifolds and study…
The parameter dependence of the various attractive solutions of the three variable nonlinear Lorenz model equations for thermal convection in Rayleigh-B\'enard flow is studied. Its bifurcation structure has commonly been investigated as a…
We determine the Lyapunov spectrum and stable manifolds of some stochastic flows on the Poincar\'e group associated to Dudley's relativistic processes.
The well-known paradox of linear stability for the some bounded shear flows is not solved up to now and is bypassed on the basis of the non-linear mechanisms consideration. We prove that it is arising only due to an idealized assumption of…
Roll-wave trains constitutes a well-known two-phase flow regime in pipes. There exists a one-parameter family of steady roll-wave train solutions, provided the flow conditions are within the roll-wave range. This means that wave train…
We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an…
We study the dynamics of a two-planet system, which evolves being in a $1/1$ mean motion resonance (co-orbital motion) with non-zero mutual inclination. In particular, we examine the existence of bifurcations of periodic orbits from the…
We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…
A universal theory of linear instabilities in swirling flows, occurring in both natural settings and industrial applications, is formulated. The theory encompasses a wide range of open and confined flows, including spiral isothermal flows…
This paper deals with stability of a certain class of fractional order linear and nonlinear systems. The stability is investigated in the time domain and the frequency domain. The general stability conditions and several illustrative…
We study Poincar\'e series associated to a finite collection of divisors on i. a finite graph and ii. a certain family of metric graphs called chain of loops. Our main results are proofs of rationality of the Poincar\'e series and…
We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not…
We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear…