Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits
نویسندگان
چکیده
A new expansion scheme to evaluate the eigenvalues of the generalized evolution operator (Frobenius-Perron operator) Hq relevant to the fluctuation spectrum and poles of the order-q power spectrum is proposed. The “partition function” is computed in terms of unstable periodic orbits and then used in a finite pole approximation of the continued fraction expansion for the evolution operator. A solvable example is presented and the approximate and exact results are compared; good agreement is found.
منابع مشابه
Continued Fractions Hierarchy of Rotation Numbers in Planar Dynamics
Global bifurcations such as crises of attractors, explosions of chaotic saddles, and metamorphoses of basin boundaries play a crucial role in understanding the dynamical evolution of physical systems. Global bifurcations in dissipative planar maps are typically caused by collisions of invariant manifolds of periodic orbits, whose dynamical behaviors are described by rotation numbers. We show th...
متن کاملTrace formulas 16
Dynamics is posed in terms of local equations, but the ergodic averages require global information. How can we use a local description of a flow to learn something about the global behavior? We have given a quick sketch of this program in Sections 1.5 and 1.6; now we redo the same material in greater depth. In Chapter 15 we have related global averages to the eigenvalues of appropriate evolutio...
متن کاملOn the real quadratic fields with certain continued fraction expansions and fundamental units
The purpose of this paper is to investigate the real quadratic number fields $Q(sqrt{d})$ which contain the specific form of the continued fractions expansions of integral basis element where $dequiv 2,3( mod 4)$ is a square free positive integer. Besides, the present paper deals with determining the fundamental unit$$epsilon _{d}=left(t_d+u_dsqrt{d}right) 2left.right > 1$$and $n_d$ and $m_d...
متن کاملPeriodic orbit spectrum in terms of Ruelle-Pollicott resonances.
Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g., a trajectory "p" returns to its initial conditions after some fixed time tau(p). Our aim is to investigate the spectrum [tau(1),tau(2), ...] of periods of the periodic orbits. An explicit formula for the density rho(tau)= Sigma(p)delta(tau-tau(p)) is derived in terms of the eigenvalues ...
متن کاملEigenfunction expansion in the singular case for q-Sturm-Liouville operators
In this work, we prove the existence of a spectral function for singular q-Sturm-Liouville operator. Further, we establish a Parseval equality and expansion formula in eigenfunctions by terms of the spectral function.
متن کامل