Paper's Title:

Alternative Proofs for the Positivity of a Function Involving the Product of Two Digamma Functions

Author(s):

Ye Shuang, Dongkyu Lim* and Feng Qi

College of Mathematical Sciences,
Inner Mongolia Minzu University,
Tongliao 028043, Inner Mongolia,
China.
E-mail: shuangye152300@sina.com
URL: https://orcid.org/0000-0002-1991-4828

 Department of Mathematics Education,
Gyeongkuk National University,
Andong 36729,
Republic of Korea.
E-mail:  dklim@gknu.ac.kr
URL: https://orcid.org/0000-0002-0928-8480

School of Mathematics and Physics,
Hulunbuir University,
Hulunbuir 021008, Inner Mongolia,
China.
17709 Sabal Court, University Village,
Dallas, TX 75252-8025,
USA.
E-mail: qifeng618@gmail.com
URL: https://orcid.org/0000-0001-6239-2968
 

Abstract:

Let ψ(x) be the digamma function, that is, the logarithmic derivative of the classical Euler's gamma function Γ(x). In this paper, the authors alternatively prove the positivity of the function

Paper's Title:

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

Author(s):

B. Deruni¹, A. S. Hacinliyan^1,2, E. Kandiran³, A. C. Keles², S. Kaouache⁴, M.-S. Abdelouahab⁴, N.-E. Hamri⁴

¹Department of Physics,
University of Yeditepe,
Turkey.

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

³Department of Software Development,
University of Yeditepe,
Turkey.

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

E-mail: berc890@gmail.com
ahacinliyan@yeditepe.edu.tr
engin.kandiran@yeditepe.edu.tr
cihan.keles@yeditepe.edu.tr
s.kaouache@centr-univ-mila.dz
medsalah3@yahoo.fr
n.hamri@centre-univ-mila.dz

Abstract:

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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