We study the origin and bifurcations of typical classes of unstable periodic orbits in a jet flow that was introduced before as a kinematic model of chaotic advection, transport and mixing of passive scalars in meandering oceanic and atmospheric currents. A method to detect and locate the unstable periodic orbits and classify them by the origin and bifurcations is developed. We consider in detail period-1 and period-4 orbits playing an important role in chaotic advection. We introduce five classes of period-4 orbits: western and eastern ballistic ones, whose origin is associated with ballistic resonances of the fourth order, rotational ones, associated with rotational resonances of the second and fourth orders, and rotational-ballistic ones associated with a rotational-ballistic resonance. It is a new kind of nonlinear resonances that may occur in chaotic flow with jets and/or circulation cells. Varying the perturbation amplitude, we track out the origin and bifurcations of the orbits for each class.
Deep Dive into Unstable periodic orbits in a chaotic meandering jet flow.
We study the origin and bifurcations of typical classes of unstable periodic orbits in a jet flow that was introduced before as a kinematic model of chaotic advection, transport and mixing of passive scalars in meandering oceanic and atmospheric currents. A method to detect and locate the unstable periodic orbits and classify them by the origin and bifurcations is developed. We consider in detail period-1 and period-4 orbits playing an important role in chaotic advection. We introduce five classes of period-4 orbits: western and eastern ballistic ones, whose origin is associated with ballistic resonances of the fourth order, rotational ones, associated with rotational resonances of the second and fourth orders, and rotational-ballistic ones associated with a rotational-ballistic resonance. It is a new kind of nonlinear resonances that may occur in chaotic flow with jets and/or circulation cells. Varying the perturbation amplitude, we track out the origin and bifurcations of the orbits
It is well known that a dynamical system is chaotic if it displays sensitivity to initial conditions, has a dense orbit and a dense set of periodic orbits (see, for example, [1]). Periodic orbits play an important role in organizing dynamical chaos both in Hamiltonian and dissipative systems. Stable periodic orbits (SPO) organize a regular motion inside islands of stability in the phase space. Unstable periodic orbits (UPO) form a skeleton around which chaotic dynamics is organized. The motion nearby an UPO is governed by its stable and unstable manifolds. Owing to the density property, the UPOs influence even the asymptotic dynamics. Order and disorder in a chaotic regime are produced eventually by an interplay between sensitivity to initial conditions and regularity of the periodic motion.
In the present paper we study origin and bifurcations of typical classes of the UPOs in a twodimensional incompressible flow that has been introduced and analyzed in Refs. [2,3,4,5,6] as a toy kinematic model of transport and mixing of passive particles in meandering jet currents in the ocean, like the Gulf Stream, Kuroshio and other main oceanic currents, and in the atmospheric currents like zonal jets with propagating Rossby waves [12,14,16]. The equations of motion of passive particles advected by any incompressible planar flow are known to have a Hamiltonian form dx dt = u(x, y, t) = -∂Ψ ∂y ,
where the streamfunction Ψ plays the role of a Hamiltonian, and the particle’s coordinates x and y are canonically conjugated variables. The phase space of Eqs. 1 is a physical space for advected particles. If the velocity field, u = u(x, y) and v = v(x, y), is stationary, then fluid particles move along streamlines, and the motion is completely regular with any Eulerian stationary field whatever its complexity. A time-periodic velocity field, u(x, y, t) = u(x, y, t+T ) and v(x, y, t) = v(x, y, t+T ), can produce chaotic particle’s trajectories, the phenomenon known as “chaotic advection” [7,8]. Chaotic advection of water (air) masses along with their properties in geophysical jets is a topic of great interest in the last decade (for recent reviews on chaotic advection, transport and mixing in the ocean and atmosphere see [9] and [10], respectively). Among the variety of kinematic and dynamic models of shear flows, one of the simplest ones is a Bickley jet with the velocity profile ∼ sech 2 y and a running wave imposed. The phase portrait of such a flow in the frame, moving with the phase velocity of the running wave, is shown in Fig. 1. The flow consists of three distinct regions, the eastward jet (J), the circulations (C) and the westward peripheral currents (P) to the north and south from the jet, separated from each other by the northern and southern ∞-like separatrices. A simple periodic modulation of the wave’s amplitude breaks up these separatrices, produces stochastic layers in place of them, and chaotic mixing and transport of passive particles may occur.
In the recent papers [4,6] we have studied statistical properties of chaotic mixing and transport in such a time-periodic meandering Bickley-jet current and explained some of them by the presence of dynamical traps in the phase space, singular zones in the stochastic layers where particles may spend arbitrary long but finite time [11]. We identified rotational-islands traps around the boundaries of rotational islands, ballistic-islands traps, around the boundaries of ballistic islands, and saddle traps associated with stable manifolds of periodic saddle trajectories [6].
A further insight into chaotic advection in the chosen model flow and in other jet flows is required in order to find a connection between dynamical and topological properties of particle’s trajectories forming a complex picture of chaotic mixing. The aim of this paper is to study in detail origin, properties and bifurcations of typical UPOs in the flow considered in Refs. [4,6]. In Sec. 2 we introduce the model streamfunction and advection equations and present a numerical method for locating the UPOs of different periods in chaotic dynamical systems. The method is based on computing a distance d between the positions, x(t 0 ), y(t 0 ), of a chosen particle at the moments of time t 0 and t 0 + mT 0 (where T 0 is a period of the perturbation and m = 1, 2, . . . ), finding local minima of the distance function with given values of m and analyzing them to locate the period-m UPOs for which d(x(t 0 ), y(t 0 )) = 0. In Sec. 3 we analyze a saddle orbit (SO) by linearizing the advection equations and study its metamorphoses varying the perturbation amplitude ε. In Sec. 4 we apply the method to study the origin, properties and bifurcations of the period-4 UPOs. We chose namely that period because for sufficiently large values of ε there are no visible period-4 resonances in the phase space. It is easy to locate the UPOs and SPOs of visible resonances on Poincaré sections, but it is not a trivial job to d
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