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Dynamical chaos in the problem of magnetic jet collimation
O. Yu. Tsupko1;2, G. S. Bisnovatyi-Kogan1;2, A. I. Neishtadt1;3, Z. F. Seidov, and Yu. M. Krivosheyev1
Space Research Institute of Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117997, Russia National Research Nuclear University MEPhI, Kashirskoe Shosse 31, Moscow 115409, Russia 3Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK
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Many quasars and active galactic nuclei are connected with long thin collimated outbursts ­ jets. When observed with high angular resolution, these jets show structure with bright knots separated by relatively dark regions. A mechanism of collimation of such jets is still questionable. Magnetic collimation of jets was first considered by Bisnovatyi-Kogan, Komberg & Fridman (1969). In the paper of Bisnovatyi-Kogan (2007) magnetic collimation resulting from torsional oscillations of a cylinder with elongated magnetic field and periodically distributed initial rotation around the cylinder axis was considered. The stabilizing azimuthal magnetic field is created here by torsional oscillations.


Magnetic collimation is connected with torsional oscillations of a cylinder with elongated magnetic field.

Bisnovatyi-Kogan, MNRAS 376, 457 (2007)

We suggest that matter in the jet is rotating, and different parts of the jet rotate in different directions: we consider a cylinder with a periodically distributed initial rotation around the cylinder axis. Such distribution of rotational velocity produces azimuthal magnetic field, which prevents the disruption of the jet. The stabilizing azimuthal magnetic field is created by torsional oscillations. The jet is represented by a periodical, or quasiperiodical structure along the axis, its radius oscillates with time only along the axis. Space and time periods of oscillations depend on the conditions of jet formation: the length-scale, the amplitude of the rotational velocity, and the strength of the magnetic field.
Cylinder undergoes magneto-torsional oscillations. Such oscillations produce a toroidal field, which prevents radial expansion. There is therefore a competition between the induced toroidal field, compressing the cylinder in the radial direction, and the gas pressure, together with the field along the cylinder axis (poloidal), tending to increase its radius. During magneto-torsional oscillations there are phases when either compression or expansion force prevails, and, depending on the input parameters, there are three possible kinds of behaviour of such a cylinder that has negligible self-gravity.


Three possibilities
1. The oscillation amplitude is low, so the cylinder suffers unlimited expansion (no confinement). 2. The oscillation amplitude is high, so the pinch action of the toroidal field destroys the cylinder and leads to the formation of separated blobs. 3. The oscillation amplitude is moderate, so the cylinder, in absence of any damping, survives for an unlimited time, and its parameters (radius, density, magnetic field, etc.) change periodically, or quasi-periodically, or chaotically in time. This case is the most interesting because it is a stable situation.
Two-dimensional non-stationary MHD calculations are needed to solve the problem numerically. Solution of the MHD equations could give, in principle, the answer about the reliability of the above scenario. It is reasonable, however, to try to find a simple approximation with which to obtain a qualitative confirmation of this scenario, and to make a rough estimation of the parameters leading to the alternative regimes. Here we construct a very simplified model of this phenomenon, which, nonetheless, permits us to confirm the reality of such stabilization, to estimate the range of parameters at which it takes place, and to establish the connection between the time and space scales, the magnetic field strength, and the amplitude of the rotational velocity. Bisnovatyi-Kogan, MNRAS 376, 457 (2007)


Approximate simplified model is developed in Bisnovatyi-Kogan, MNRAS 376, 457 (2007). Ordinary differential equation is derived, and solved numerically, what gives a possibility to estimate quantitatively the range of parameters where jets may be stabilized by torsional oscillations.

In non-dimensional variables differential equations have a form (\tau ­ time; y - radius; z - radial velocity):

The ordinary differential equation under consideration is a non-linear nonautonomous time-periodic second order equation with a singularity in the right hand side. The equation contains one dimensionless parameter D, which summarizes the information about the magnetic field, amplitude and frequency of oscillations, radius of the jet, its spatial period along the jet axis, and sound speed of jet matter.



(\tau ­ time; y - radius; z - radial velocity

Here we investigate analytically and numerically the structure of the phase space of this equation, which has a very peculiar character and contains chaotic solutions as well as quasi-periodic and periodic regular solutions. Numerical solutions in Bisnovatyi-Kogan, MNRAS 376, 457 (2007) were done only for y(0)=1, z(0)=0 at different D. When y(0) is different from 1, there is a larger variety of solutions: periodic, regular and chaotic.


Numerical solution
1. At D = 1.5 and 2 there is no confinement, and radius grows to infinity after several lowamplitude oscillations

Solutions only for y(0)=1, z(0)=0

Bisnovatyi-Kogan, MNRAS 376, 457 (2007)


2. With growing of D the amplitude of oscillations increase, and at D=2.1 radius is not growing to infinity, but is oscillating around some average value, forming rather complicated curve. The oscillations are regular or chaotic, depending on the parameters.


3. At D= 2.28 and larger the radius finally goes to zero with time, but with different behavior, depending on D. At D between 2.28 and 2.9 the dependence of the radius y with time may be very complicated, consisting of low-amplitude and large-amplitude oscillations, which finally lead to zero. The time at which radius becomes zero depends on D in rather peculiar way, and may happen at t< 100, like at D=2.4, 2.6; or goes trough very large radius, and returned back to zero value at very large time t » 10^7 at D=2.5 . Starting from D = 3 and larger the solution becomes very simple, and radius goes to zero at t < 2:5, before the right side of the second equation returned to the positive value.


Investigation of phase space of equation, Development of chaos, Poincare sections
G. S. Bisnovatyi-Kogan, A. I. Neishtadt, Z. F. Seidov, O. Yu. Tsupko, and Yu. M. Krivosheyev
(2011, in preparation)

In the current work we investigate the solutions of equation for different values of D and different initial radii y0, mainly for moderate D's. Solutions of principal interest are those that do not go to 0 nor to infinity. Such solutions correspond to stabilized jets.





TRANSITION TO CHAOS VIA CASCADE OF PERIOD DOUBLINGS

With increasing D the stable periodic solution loses its stability via period doubling bifurcation. The first period doubling occurs at D1 = 2.38101. At this value D1 the solution with period becomes unstable, and a stable periodic solution of period 2 appears.




The Doublings Points
The following doubling points found by the Poincarґe section construction are:

AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations (Doedel & Oldeman (2009), http://cmvl.cs.concordia.ca/auto/).

The values D1, D2, D3, D4 have also been obtained with the help of


In the problem of period doubling the limiting constant q for values [Dn ­ Dn-1]/[Dn+1 ­ Dn] is usually considered. For large n the constant q equals to Feigenbaum constant F=4.6692... for dissipative systems, see Feigenbaum (1980), and FH=8.721... for Hamiltonian systems, see Greene et al. (1981).

Feigenbaum sequence for the parameter D to reach the chaos
We conclude that this behavior agrees with the expected one for the Hamiltonian systems.


Conclusions
1. Jets can be magnetically collimated by large scale magnetic field by torsion oscillations, which may be periodic, regular or chaotic. 2. Equation describing magnetic collimation has complicated phase space. Chaos is reached via the cascade of period doublings. behavior agrees with the expected one (Feigenbaum sequence) for the Hamiltonian systems.
Publications: 1. Bisnovatyi-Kogan, MNRAS 376, 457 (2007) 2. G. S. Bisnovatyi-Kogan, A. I. Neishtadt, Z. F. Seidov, O. Yu. Tsupko, and Yu. M. Krivosheyev (2011, in preparation)