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13: Paper Source PDF document

Paper's Title:

An Integration Technique for Evaluating Quadratic Harmonic Sums


J. M. Campbell and K.-W. Chen

Department of Mathematics and Statistics,
York University, 4700 Keele St, Toronto,
ON M3J 1P3,

Department of Mathematics, University of Taipei,
  No. 1, Ai-Guo West Road,
Taipei 10048, Taiwan.


The modified Abel lemma on summation by parts has been applied in many ways recently to determine closed-form evaluations for infinite series involving generalized harmonic numbers with an upper parameter of two. We build upon such results using an integration technique that we apply to ``convert'' a given evaluation for such a series into an evaluation for a corresponding series involving squared harmonic numbers.

3: Paper Source PDF document

Paper's Title:

Coexisting Attractors and Bubbling Route to Chaos in Modified Coupled Duffing Oscillators


B. Deruni1, A. S. Hacinliyan1,2, E. Kandiran3, A. C. Keles2, S. Kaouache4, M.-S. Abdelouahab4, N.-E. Hamri4

1Department of Physics,
University of Yeditepe,

2Department of Information Systems and Technologies,
University of Yeditepe,

3Department of Software Development,
University of Yeditepe,

4Laboratory of Mathematics and their interactions,
University Center of Abdelhafid Boussouf,
Mila 43000,



In this article dynamical behavior of coupled Duffing oscillators is analyzed under a small modification. The oscillators have cubic damping instead of linear one. Although single duffing oscillator has complex dynamics, coupled duffing systems possess a much more complex structure. The dynamical behavior of the system is investigated both numerically and analytically. Numerical results indicate that the system has double scroll attractor with suitable parameter values. On the other hand, bifurcation diagrams illustrate rich behavior of the system, and it is seen that, system enters into chaos with different routes. Beside classical bifurcations, bubbling route to chaos is observed for suitable parameter settings. On the other hand, Multistability of the system is indicated with the coexisting attractors, such that under same parameter setting the system shows different periodic and chaotic attractors. Moreover, chaotic synchronization of coupled oscillators is illustrated in final section.

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