| dc.description.abstract | This thesis consists of two parts. The first part describes a new method of computing the Lyapunov exponents of continuous and discrete dynamical systems. Chaos is observed in a large class of systems and hence the problem of detecting and quantifying it becomes important. The hallmark of chaotic systems is the exponential separation of initially nearby trajectories in time, called sensitivity to initial conditions. Due to this exponential sensitivity to initial conditions, even very small round-off errors in the solution grow rapidly with time. Consequently, after a finite time, the solution is totally different from what it would have been in the absence of these errors, leading to a total loss of predictability. Though there are many methods to detect chaos, the set of Lyapunov exponents is the most practical diagnostic tool available. These exponents quantify the average exponential rates of divergence or convergence of initially nearby orbits in the phase space. Since nearby orbits correspond to nearly identical states, exponential orbital divergence implies that in systems, whose initial differences we may not be able to resolve, predictability is lost. Any system containing at least one positive Lyapunov exponent is defined to be chaotic, with the magnitude of the exponent reflecting the time scale on which system dynamics become unpredictable.
For systems, whose equations of motions are explicitly known, there exist several methods for computing Lyapunov exponents. All these existing methods are based on either QR or SVD (singular value decomposition) methods. In case of the QR method, the tangent map is decomposed into an orthogonal matrix Q and an upper triangular matrix R, while in the SVD method, it is written as a product of an orthogonal matrix, a diagonal matrix and another orthogonal matrix. All the existing methods suffer from one or more of the following disadvantages:
They require frequent renormalization to combat exponential growth of the separation vectors between the fiducial and nearby trajectories and reorthogonalization to overcome the exponential collapse of initially orthogonal separation vectors onto the direction of maximal growth.
The continuous versions of these methods suffer from the additional disadvantage of being unable to compute the partial spectrum, using fewer number of equations than required to calculate the complete spectrum.
The SVD method breaks down if the system has a degenerate Lyapunov spectrum.
In this thesis, we propose a new method, which does away with all the disadvantages encountered in the existing methods. This method is again based on the QR decomposition, but instead of working directly with the matrix elements of the orthogonal matrix, we use the group theoretical representation of the orthogonal matrix. The use of these representations analytically obviates the need for rescaling and reorthogonalization. This method also does away with the other shortcomings of the existing methods and has the following advantages:
A partial Lyapunov spectrum can be computed using a fewer number of equations as compared to the computation of the full spectrum.
There is no difficulty in evaluating degenerate Lyapunov spectra unlike the SVD method.
The equations are straightforward to generalize to higher dimensions, and the method uses the minimal set of dynamical variables.
Since our method is based on exact differential equations for the Lyapunov exponents, global invariances of the Lyapunov spectrum are preserved.
All the above statements are analytically substantiated with proofs in this thesis. We have illustrated the working of this method with the help of a few examples. We have also adapted our method to discrete dynamical systems, while retaining all the above advantages.
The group theoretical representation of the special orthogonal matrix is extensively used in our method. It involves a lot of cumbersome matrix multiplications for higher dimensions. In this thesis, we have obtained the expression for a general element of an SO(n) matrix, which obviates the need for matrix multiplications. Matrix representations of the SO(n) group have played an important role in theoretical physics. They continue to be used in many fields even today. They also play a crucial role in deriving the expressions for the Lyapunov exponents. In this thesis, we have demonstrated that expressions for all elements can be derived from a single matrix element. We have derived the elements of an SO(3) matrix as an example of an application of this result.
The second part of this thesis consists of the work done on the dynamical moment invariants of nonlinear Hamiltonian systems using the normal form technique. We study the distributions of particles being transported through a nonlinear Hamiltonian system. Using the normal form technique, a procedure to obtain invariant functions of moments of the distribution is given. These functions are invariant for the given Hamiltonian system and are called dynamic moment invariants. This technique is then used to obtain dynamic moment invariants for the nonlinear pendulum Hamiltonian. The dynamical moment invariants of a Hamiltonian system serve as a measure of the nonlinearity of the system i.e., if their magnitude is large, the system is more nonlinear. They can also be used as a check for the correctness of moment codes (algorithms, which numerically integrate the evolution of a distribution, using the moments of the distribution). | |
