Short communication
Bifurcation continuation, chaos and chaos control in nonlinear Bloch system

https://doi.org/10.1016/j.cnsns.2007.03.009Get rights and content

Abstract

A detailed analysis is undertaken to explore the stability and bifurcation pattern of the nonlinear Bloch equation known to govern the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance. After the initial analysis of the parameter space and stability region identification, we utilize the MATCONT package to analyze the detailed bifurcation scenario as the two important physical parameters γ (the normalized gain) and c (the phase of the feedback field) are varied. A variety of patterns are revealed not studied ever before. Next we explore the structure of the chaotic attractor and how the identification of unstable periodic orbit (UPO) can be utilized to control the onset of chaos.

Introduction

The dynamics of an ensemble of spins in an external magnetic field is of utmost importance in the understanding various phenomena related to the extremely important field for nuclear magnetic resonance (NMR). The actual physical process can be described as a combination of the precession of the spins about a magnetic field and damping of transverse and longitudinal component of magnetization with different relaxation time Γ1 and Γ2. Manifestations of nonlinear spin dynamics could be observed due to the presence of an additional field which is proportional to the components of the magnetization. The dipolar magnetizing field could be shown to give rise to multiple echoes in liquid helium [1] and in water at high magnetic field [2]. Moreover, the effect of radiation damping or demagnetizing field [3], [4], [5] can result in the existence of pseudo-multiquanta peaks. A good description for the whole phenomena is given by Bloch equation, whose preliminary analysis in very special cases were reported by Abergel et al. [6], who studied some special situation when the relaxation time was large or small in an asymptotic way. Though they obtained the chaotic attractor in the general case, the over all scenario and the various channel of bifurcation was not obtained. Here in this communication we have studied in detail the stability zones in parameter space and the different route of continuation of the associated bifurcation with the help of the software package MATCONT [7], [8], [9], revealing a rich structure of dynamical transitions. In the next part of our paper have studied the structure of the unstable periodic orbits and have devised a method to control the chaos so generated.

The work of OGY [10] and Pecora and Carroll [11] led to wide applications outside the traditional scope of chaos and nonlinear dynamics. The unified study of chaos control and synchronization was carried out in [12], [13]. Different approaches have been suggested for such procedure [14], [15]. Incidentally Yu et al. [16] proposed a method for controlling chaos in the form of special nonlinear feedback. The validity of this method based on the stability criterion of linear system and can be called stability criterion method (SC method). The construction of a nonlinear form of limit continuous perturbation feedback by a suitable separation of the system in the SC method does not change the form of the desired UPO. The closed return pair technique [17] is utilized to estimate the desired periodic orbit chosen from numerous UPO’s embedded within a chaotic attractor. The advantage of this method is that the effect of the control can be generalized directly without calculation of the maximal lyapunov exponent of the UPO using the linearization of the system. The method has been used by researchers in the control of Rössler system, chaotic altitude motion of a spacecraft and the control of two coupled Duffing oscillators.

In this communication, the local stability analysis of the nonlinear Bloch equations in a particular parameter regime has been discussed and then the different route of continuation of the associated bifurcation analysis is investigated. Finally, the chaotic scenario of the system is controlled using the stability criterion method which involves the construction of input perturbation function.

Section snippets

Formulation

The dynamics of an ensemble of spins usually described by the nonlinear Bloch equation is very important for the understanding of the underlying physical process of nuclear magnetic resonance. The basic process can be viewed as the combination of a precession about a magnetic field and of a relaxation process, which gives rise to the damping of the transverse component of the magnetization with a different time constant. Introduction of an externally generated feedback field was first proposed

Bifurcation and continuation

The main goal of the previous section was to qualitatively characterize the way the stationary solution of the magnetization is approached at long times. Our next goal is to study the pattern of bifurcation that takes place as we vary the parameters γ and c. This is actually done by studying the change in the eigenvalue of the Jacobian matrix and also following the continuation algorithm. To start with we consider a set of fixed point solution, x0 = 0.2761, y0 = 0.0236 and z0 = 0.8569, corresponding

Onset of chaos and chaos control

From the detailed picture of bifurcation given above it is now very easy to see that certain values of γ and c actually leads to the chaotic attractor. The form of attractor is given in Fig. 9a and b. The two attractors occurs for two different sets of values:γ=35.0,δ=-1.26,c=0.173,Γ1=5.0,Γ2=2.5andγ=10.0,δ=1.26,c=0.7764,Γ1=0.5,Γ2=0.25.The presence of chaotic attractor indicates the presence of numerous unstable periodic orbits. Such type of chaotic behaviour can be controlled by different

Conclusion

In our above analysis we have explored the detailed bifurcation scenario of the set of nonlinear equations describing the spin dynamics in an external magnetic field (useful in the elaborate study of NMR) when two physical parameters of the system γ (the normalized gain) and c (the phase of the feedback field) are varied. The interesting outcome is the occurrence of various kinds of bifurcation points as the process of continuation takes place, giving an insight into the formation and route to

Acknowledgement

One of the authors D. Ghosh is thankful to CSIR, Govt. of India, for a junior research fellowship.

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