Scholar iON
Academic Synthesis
The collection of scholarly papers explores diverse yet interrelated phenomena within nonlinear systems, focusing on statistical mechanics, random matrix theory, quantum graphs, and dynamical systems. Celani et al. (2005) investigate polymer dynamics in shear flow, using numerical methods to reveal alignment and elongation behaviors consistent with experimental observations, highlighting the role of thermal fluctuations. Fyodorov and Sommers (2000) analyze eigenvalue distributions in random contractions, providing a comprehensive statistical framework for systems with broken time-reversal invariance. Berkolaiko et al. (2001) extend periodic-orbit theory, elucidating corrections in quantum graph form factors and aligning with random-matrix theory predictions. San-MartΓn (2005) examines universal scaling in bifurcation cascades, merging concepts of intermittency and bifurcation, thus enriching the understanding of complex dynamical transitions. Collectively, these studies contribute significantly to the theoretical and computational understanding of nonlinear dynamics and complex systems, bridging gaps between theoretical predictions and empirical findings.
We study the dynamics of a single polymer subject to thermal fluctuations in a linear shear flow. The polymer is modeled as a finitely extendable nonlinear elastic FENE dumbbell. Both orientation and elongation dynamics are investigated numerically as a function of the shear strength, by means of a new efficient integration algorithm. The results are in agreement with recent experiments.
Random contractions (sub-unitary random matrices) appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. We analyze statistical properties of complex eigenvalues of generic $N\times N$ random matrices $\hat{A}$ of such a type, corresponding to systems with broken time-reversal invariance. Deviations from unitarity are characterized by rank $M\le N$ and a set of eigenvalues $0<T_i\le 1, i=1,...,M$ of the matrix $\hat{T}=\hat{\bf 1}-\hat{A}^{\dagger}\hat{A}$. We solve the problem completely by deriving the joint probability density of $N$ complex eigenvalues and calculating all $n-$ point correlation functions. In the limit $N>>M,n$ the correlation functions acquire the universal form found earlier for weakly non-Hermitian random matrices.
Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, $K(Ο)$, of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution $-2Ο^2$ to the form factor which agrees with random-matrix theory.
The presence of saddle-node bifurcation cascade in the logistic equation is associated with an intermittency cascade; in a similar way as a saddle-node bifurcation is associated with an intermittency. We merge the concepts of bifurcation cascade and intermittency. The mathematical tools necessary for this process will describe the structure of the Myrberg-Feigenbaum point.
Using standard definitions of chaos (as positive Kolmogorov-Sinai entropy) and diffusion (that multiple time distribution functions are Gaussian), we show numerically that both chaotic and nonchaotic systems exhibit diffusion, and hence that there is no direct logical connection between the two properties. This extends a previous result for two time distribution functions.
Difference control schemes for controlling unstable fixed points become important if the exact position of the fixed point is unavailable or moving due to drifting parameters. We propose a memory difference control method for stabilization of a priori unknown unstable fixed points by introducing a memory term. If the amplitude of the control applied in the previous time step is added to the present control signal, fixed points with arbitrary Lyapunov numbers can be controlled. This method is also extended to compensate arbitrary time steps of measurement delay. We show that our method stabilizes orbits of the Chua circuit where ordinary difference control fails.