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Journal articles on the topic 'Coupled harmonic oscillator'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Dattoli, G., A. Torre, S. Lorenzutta, and G. Maino. "Coupled harmonic oscillators, generalized harmonic-oscillator eigenstates and coherent states." Il Nuovo Cimento B Series 11 111, no. 7 (July 1996): 811–23. http://dx.doi.org/10.1007/bf02749013.

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2

Wang, Shijiao, Xiao San Ma, and Mu-Tian Cheng. "Multipartite Entanglement Generation in a Structured Environment." Entropy 22, no. 2 (February 7, 2020): 191. http://dx.doi.org/10.3390/e22020191.

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In this paper, we investigate the entanglement generation of n-qubit states in a model consisting of n independent qubits, each coupled to a harmonic oscillator which is in turn coupled to a bath of N additional harmonic oscillators with nearest-neighbor coupling. With analysis, we can find that the steady multipartite entanglement with different values can be generated after a long-time evolution for different sizes of the quantum system. Under weak coupling between the system and the harmonic oscillator, multipartite entanglement can monotonically increase from zero to a stable value. Under strong coupling, multipartite entanglement generation shows a speed-up increase accompanied by some oscillations as non-Markovian behavior. Our results imply that the strong coupling between the harmonic oscillator and the N additional harmonic oscillators, and the large size of the additional oscillators will enhance non-Markovian dynamics and make it take a very long time for the entanglement to reach a stable value. Meanwhile, the couplings between the additional harmonic oscillators and the decay rate of additional harmonic oscillators have almost no effect on the multipartite entanglement generation. Finally, the entanglement generation of the additional harmonic oscillators is also discussed.

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3

Dudinetc, I. V., and V. I. Man’ko. "Quantum correlations for two coupled oscillators interacting with two heat baths." Canadian Journal of Physics 98, no. 4 (April 2020): 327–31. http://dx.doi.org/10.1139/cjp-2019-0067.

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We study a system of two coupled oscillators (A oscillators), each of which linearly interact with their own heat bath consisting of a set of independent harmonic oscillators (B oscillators). The initial state of the A oscillator is taken to be coherent while the B oscillator is in a thermal state. We analyze the time-dependent state of the A oscillator, which is a two-mode Gaussian state. By making use of Simon’s separability criterion, we show that this state is separable for all times. We consider the equilibrium state of the A oscillator in detail and calculate its Wigner function.

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4

LAWANDE, S. V., and Q. V. LAWANDE. "PATH INTEGRAL DERIVATION OF AN EXACT MASTER EQUATION." Modern Physics Letters B 09, no. 02 (January 20, 1995): 87–94. http://dx.doi.org/10.1142/s0217984995000097.

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The Feynman propagator in coherent states representation is obtained for a system of a single harmonic oscillator coupled to a reservoir of N oscillators. Using this propagator, an exact master equation is obtained for the evolution of the reduced density matrix for the open system of the oscillator.

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5

Chakrabarti, Barnali, and Bambi Hu. "Level correlation in coupled harmonic oscillator systems." Physics Letters A 315, no. 1-2 (August 2003): 93–100. http://dx.doi.org/10.1016/s0375-9601(03)01001-6.

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6

Merdaci, Abdeldjalil, Ahmed Jellal, Ayman Al Sawalha, and Abdelhadi Bahaoui. "Purity temperature dependency for coupled harmonic oscillator." Journal of Statistical Mechanics: Theory and Experiment 2018, no. 9 (September 6, 2018): 093101. http://dx.doi.org/10.1088/1742-5468/aad19b.

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7

Maidanik, G., and K. J. Becker. "Noise control of a master harmonic oscillator coupled to a set of satellite harmonic oscillators." Journal of the Acoustical Society of America 104, no. 5 (November 1998): 2628–37. http://dx.doi.org/10.1121/1.423846.

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8

Tung, Mingwhei, Elia Eschenazi, and Jian-Min Yuan. "Dynamics of a morse oscillator coupled to a bath of harmonic oscillators." Chemical Physics Letters 115, no. 4-5 (April 1985): 405–10. http://dx.doi.org/10.1016/0009-2614(85)85158-7.

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9

Burrows, B. L., M. Cohen, and Tova Feldmann. "Coupled harmonic oscillator systems: Improved algebraic decoupling approach." International Journal of Quantum Chemistry 92, no. 4 (2003): 345–54. http://dx.doi.org/10.1002/qua.10521.

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10

Maidanik, G., and K. J. Becker. "Various loss factors of a master harmonic oscillator coupled to a number of satellite harmonic oscillators." Journal of the Acoustical Society of America 103, no. 6 (June 1998): 3184–95. http://dx.doi.org/10.1121/1.423035.

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11

Ullah, Farman, Yu Liu, Zhiqiang Li, Xiaosong Wang, Muhammad Sarfraz, and Haiying Zhang. "A Bandwidth-Enhanced Differential LC-Voltage Controlled Oscillator (LC-VCO) and Superharmonic Coupled Quadrature VCO for K-Band Applications." Electronics 7, no. 8 (July 25, 2018): 127. http://dx.doi.org/10.3390/electronics7080127.

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A novel varactor circuit exhibiting a wider tuning range and a new technique for quadrature coupling of LC-Voltage Controlled Oscillator (LC-VCO) is presented and validated on a 25 GHz oscillator. The proposed varactor circuit employs distribute-biased parallel varactors with a series inductor connected at both ends of the varactor bank to extend the tuning range of the oscillator. Similarly, the quadrature coupling is accomplished by employing the 2nd harmonic, explicitly generated in the stand-alone free-running differential oscillator using frequency doubler. As an example, the Differential VCO (DVCO) is tunable between 20 GHz and 31 GHz and exhibits the best Phase Noise (PN) of −100 dBc/Hz at 1 MHz offset frequency. Similarly, the Quadrature VCO (QVCO) covers 42% tuning bandwidth around 25 GHz oscillation frequency, which is significantly wider than other state-of-the-art VCOs at comparable frequencies. In addition, all the oscillators are designed in class-C to further improve their performances both in term of low power and low phase noise. The presented oscillators are designed using high-performance SiGe HBTs of the GlobalFoundries (GFs) 130 nm SiGe BiCMOS 8HP process. The presented DVCO and QVCO draw currents of approximately 10 mA and 21 mA, respectively from a 1.2 V supply.

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12

Manevitch, L. I., A. S. Kovaleva, and E. L. Manevitch. "Limiting Phase Trajectories and Resonance Energy Transfer in a System of Two Coupled Oscillators." Mathematical Problems in Engineering 2010 (2010): 1–24. http://dx.doi.org/10.1155/2010/760479.

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We study a problem of energy exchange in a system of two coupled oscillators subject to 1 : 1 resonance. Our results exploit the concept of limiting phase trajectories (LPTs). The LPT, associated with full energy transfer, is, in certain sense, an alternative to nonlinear normal modes characterized by conservation of energy. We consider two benchmark examples. As a first example, we construct an LPT and examine the convergence to stationary oscillations for a Duffing oscillator subjected to resonance harmonic excitation. As a second example, we treat resonance oscillations in a system of two nonlinearly coupled oscillators. We demonstrate the reduction of the equations of motion to an equation of a single oscillator. It is shown that the most intense energy exchange and beating arise when motion of the equivalent oscillator is close to an LPT. Damped beating and the convergence to rest in a system with dissipation are demonstrated.

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13

MAEZAWA, Koichi, Takashi OHE, Koji KASAHARA, and Masayuki MORI. "A Third Order Harmonic Oscillator Based on Coupled Resonant Tunneling Diode Pair Oscillators." IEICE Transactions on Electronics E93-C, no. 8 (2010): 1290–94. http://dx.doi.org/10.1587/transele.e93.c.1290.

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14

Zeng, Tianhai, Zhaobin Liu, Kai Li, Feng Wang, and Bin Shao. "Outgrowth of quasi pure states in isolated coupled-harmonic-oscillator." International Journal of Modern Physics B 35, no. 05 (February 3, 2021): 2150075. http://dx.doi.org/10.1142/s0217979221500752.

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Isolated coupled-harmonic-oscillator here is the system of two distinguishable particles coupled with a harmonic oscillator interaction potential. Each particle stays in a mixed state due to entanglement. However, in center-of-mass reference frame, we obtain quasi wavefunction of the first particle expressing quasi pure state by replacing the second coordinate in the total wavefunction. We discuss the similar systems with the first particle and the potential being same and the second mass changing from micro to macro one. Measured by fidelity and coherence, the quasi pure state approaches to the pure state of a usual harmonic oscillator with same mass and similar potential. It conversely shows that the latter purely superposed state in position representation and its coherence originate from those of the first particle, which are related with some neglected macro object and the interaction between them. The current results provide a possible clue to new insights into quantum states.

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15

YANG, SHI-PING, YAN LI, HONG CHANG, GANG TIAN, and GUO-YONG YUAN. "THE STUDY OF COUPLED OSCILLATORS ON CLASSICAL CHAOS AND QUANTUM CHARACTERISTIC." International Journal of Modern Physics B 20, no. 24 (September 30, 2006): 3465–75. http://dx.doi.org/10.1142/s021797920603562x.

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In this paper, the classical and quantum chaos characteristic of single electronic motion in the double quantum well with external magnetic field are studied. The system can be regarded as the linear coupling of a harmonic oscillator and a Duffing oscillator. The study shows that because of the interaction of two oscillators, the system demonstrates the characteristic of quasi-periodicity, multi-chaos coexisting attractors, chaos, super-chaos, etc., with different energy. Furthermore, as shown in the corresponding analysis of spectrum distribution statistics, the system in most energy fields demonstrates the coexisting of the integrable and non-integrable characteristic, which means that there is a close corresponding relation in classical and quantum behavior.

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16

Preyer, Norris W. "The coupled harmonic oscillator: Not just for seniors anymore." Physics Teacher 34, no. 1 (January 1996): 52–55. http://dx.doi.org/10.1119/1.2344340.

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17

Djurhuus, T., V. Krozer, J. Vidkjaer, and T. K. Johansen. "Nonlinear analysis of a cross-coupled quadrature harmonic oscillator." IEEE Transactions on Circuits and Systems I: Regular Papers 52, no. 11 (November 2005): 2276–85. http://dx.doi.org/10.1109/tcsi.2005.853586.

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18

Collado, A., F. Ramirez, A. Suarez, and J. P. Pascual. "Harmonic-balance analysis and synthesis of coupled-oscillator arrays." IEEE Microwave and Wireless Components Letters 14, no. 5 (May 2004): 192–94. http://dx.doi.org/10.1109/lmwc.2004.827863.

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19

Huang, F. H., C. K. Lin, and Y. J. Chan. "V-Band GaAs pHEMT Cross-Coupled Sub-Harmonic Oscillator." IEEE Microwave and Wireless Components Letters 16, no. 8 (August 2006): 473–75. http://dx.doi.org/10.1109/lmwc.2006.879479.

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20

Feldmann, Tova, M. Cohen, and B. L. Burrows. "Coupled harmonic oscillator systems: An elementary algebraic decoupling approach." Journal of Mathematical Physics 41, no. 9 (September 2000): 5897–909. http://dx.doi.org/10.1063/1.1288248.

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21

Williams, Cameron L., Nikhil N. Pandya, Bernhard G. Bodmann, and Donald J. Kouri. "Coupled supersymmetry and ladder structures beyond the harmonic oscillator." Molecular Physics 116, no. 19-20 (May 18, 2018): 2599–612. http://dx.doi.org/10.1080/00268976.2018.1473655.

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22

Yu, Bo, James S. Freudenberg, R. Brent Gillespie, and Richard H. Middleton. "Beyond synchronization: String instability in coupled harmonic oscillator systems." International Journal of Robust and Nonlinear Control 25, no. 15 (August 12, 2014): 2745–69. http://dx.doi.org/10.1002/rnc.3229.

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23

Blasone, M., and P. Jizba. "Quantum mechanics of the damped harmonic oscillator." Canadian Journal of Physics 80, no. 6 (June 1, 2002): 645–60. http://dx.doi.org/10.1139/p02-003.

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We quantize the system of a damped harmonic oscillator coupled to its time-reversed image, known as Bateman's dual system. By using the FeynmanHibbs method, the time-dependent quantum states of such a system are constructed entirely in the framework of the classical theory. The geometric phase is calculated and found to be proportional to the ground-state energy of the one-dimensional linear harmonic oscillator to which the two-dimensional system reduces under appropriate constraint. PACS Nos.: 03.65Ta, 03.65Vf, 03.65Ca, 03.65Fd

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24

Gallo, James Mendoza, and Bienvenido Masirin Butanas Jr. "Quantum Propagator Derivation for the Ring of Four Harmonically Coupled Oscillators." Jurnal Penelitian Fisika dan Aplikasinya (JPFA) 9, no. 2 (December 31, 2019): 92. http://dx.doi.org/10.26740/jpfa.v9n2.p92-104.

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The ring model of the coupled oscillator has enormously studied from the perspective of quantum mechanics. The research efforts on this system contribute to fully grasp the concepts of energy transport, dissipation, among others, in mesoscopic and condensed matter systems. In this research, the dynamics of the quantum propagator for the ring of oscillators was analyzed anew. White noise analysis was applied to derive the quantum mechanical propagator for a ring of four harmonically coupled oscillators. The process was done after performing four successive coordinate transformations obtaining four separated Lagrangian of a one-dimensional harmonic oscillator. Then, the individual propagator was evaluated via white noise path integration where the full propagator is expressed as the product of the individual propagators. In particular, the frequencies of the first two propagators correspond to degenerate normal mode frequencies, while the other two correspond to non-degenerate normal mode frequencies. The full propagator was expressed in its symmetric form to extract the energy spectrum and the wave function.

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25

Galve, Fernando, and Roberta Zambrini. "Energy and information propagation in a finite coupled bosonic heat bath." International Journal of Quantum Information 12, no. 07n08 (November 2014): 1560022. http://dx.doi.org/10.1142/s0219749915600229.

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The finite coupled bosonic model of reservoir introduced by Vasile et al. [Phys. Rev. A 89 (2014) 022109] to characterize non-Markovianity, is used to study the different dissipative behaviors of a harmonic oscillator coupled to it when it is in resonance, close to resonance or far detuned. We show that information and energy exchange between system and heat bath go hand in hand because phonons are the carriers of both: in resonance free propagation of excitations is achieved, and therefore pure dissipation, while when far detuned the system can only correlate with the first oscillator in the bath's chain, leading to almost unitary evolution. In the intermediate situation, we show the penetration of correlations and the formation of oscillatory (dressed state) behavior, which lies at the root of non-Markovianity.

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26

Guldberg, Annette, and Gert D. Billing. "Laser-induced dissociation of an anharmonic oscillator coupled to a set of harmonic oscillators." Chemical Physics Letters 191, no. 5 (April 1992): 455–62. http://dx.doi.org/10.1016/0009-2614(92)85408-3.

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27

Nguyen, Thanh Dat, and Jong-Phil Hong. "A 350-GHz Coupled Stack Oscillator with −0.8 dBm Output Power in 65-nm Bulk CMOS Process." Electronics 9, no. 8 (July 28, 2020): 1214. http://dx.doi.org/10.3390/electronics9081214.

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This paper presents a push-push coupled stack oscillator that achieves a high output power level at terahertz (THz) wave frequency. The proposed stack oscillator core adopts a frequency selective negative resistance topology to improve negative transconductance at the fundamental frequency and a transformer connected between gate and drain terminals of cross pair transistors to minimize the power loss at the second harmonic frequency. Next, the phases and the oscillation frequencies between the oscillator cores are locked by employing an inductor of frequency selective negative resistance topology. The proposed topology was implemented in a 65-nm bulk CMOS technology. The highest measured output power is −0.8 dBm at 353.2 GHz while dissipating 205 mW from a 2.8 V supply voltage.

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28

Hirokawa, M. "Mori′s Memory Kernel Equation for a Quantum Harmonic Oscillator Coupled to RWA-Oscillator." Annals of Physics 224, no. 2 (June 1993): 301–41. http://dx.doi.org/10.1006/aphy.1993.1048.

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29

Ebrahimi, Emad, and Sasan Naseh. "A new robust capacitively coupled second harmonic quadrature LC oscillator." Analog Integrated Circuits and Signal Processing 66, no. 2 (August 10, 2010): 269–75. http://dx.doi.org/10.1007/s10470-010-9512-6.

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30

Grimaudo, R., V. I. Man’ko, M. A. Man’ko, and A. Messina. "Dynamics of a harmonic oscillator coupled with a Glauber amplifier." Physica Scripta 95, no. 2 (December 31, 2019): 024004. http://dx.doi.org/10.1088/1402-4896/ab4305.

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31

Paganelli, S., and S. Ciuchi. "Tunnelling system coupled to a harmonic oscillator: an analytical treatment." Journal of Physics: Condensed Matter 18, no. 32 (July 31, 2006): 7669–85. http://dx.doi.org/10.1088/0953-8984/18/32/015.

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32

Billionnet, C. "On resonances for a harmonic oscillator coupled with massless bosons." Journal of Physics A: Mathematical and General 31, no. 2 (January 16, 1998): 623–38. http://dx.doi.org/10.1088/0305-4470/31/2/020.

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33

Zúñiga, José, Adolfo Bastida, and Alberto Requena. "Quantum solution of coupled harmonic oscillator systems beyond normal coordinates." Journal of Mathematical Chemistry 55, no. 10 (July 4, 2017): 1964–84. http://dx.doi.org/10.1007/s10910-017-0777-1.

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34

Billionnet, C. "Resonances Formed by Massless Bosons Coupled to a Harmonic Oscillator." Annales Henri Poincaré 2, no. 2 (April 2001): 361–76. http://dx.doi.org/10.1007/pl00001037.

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35

Giraldi, F., and F. Petruccione. "Anomalies in Strongly Coupled Harmonic Quantum Brownian Motion II." Open Systems & Information Dynamics 20, no. 04 (November 25, 2013): 1350015. http://dx.doi.org/10.1142/s1230161213500157.

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The analysis of a strongly coupled harmonic quantum Brownian motion has been performed in [1] for a special class of spectral densities obtained as a generalization of the Drude model. In the present scenario, we extend the study of the strongly coupled harmonic quantum Brownian motion to regular spectral densities that are structured as sub-Ohmic at low frequencies and arbitrarily shaped at high frequencies. The bosonic environment is initially in the vacuum state unentangled from the coherent state of the main oscillator. As a generalization of the previous results, we obtain that the long time dynamics is determined uniquely by the initial condition and the low frequency structure of the spectral density. Also in the present framework, inverse power law regressions to the asymptotics appear. The position and the momentum tend to undamped oscillations. The number of excitations relaxes to its initial value with damped oscillations enveloped in inverse power law relaxations. For the momentum and the number of excitations the inverse power law decays become arbitrarily slow in critical configurations by approaching the upper bound of the sub-Ohmic regime. The critical frequencies of the main oscillator are determined by the first negative moment of the spectral densities. Differences with respect to the weak coupling regime arise.

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36

Hassan, S. S., R. A. Alharbey, and H. Al-Zaki. "Transient spectrum of sin2-pulsed driven harmonic oscillator." Journal of Nonlinear Optical Physics & Materials 23, no. 04 (December 2014): 1450052. http://dx.doi.org/10.1142/s0218863514500520.

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Exact analytical results are derived for the coupled system of nondissipative single mode quantized harmonic oscillator (HO) and arbitrary pulse shape. Specifically, for a sin2-pulse shape, the transient fluorescent spectrum is obtained in the general case of initial coherent state |α〉 of the HO. The dominant central Lorentzian is surrounded by weak oscillations due to the larger number of sequential pulses which get amplified asymmetrically in the nonresonant case as a result of balancing and interference processes between the initial excitation (|α2|) and the strength of the exciting pulse. Further pronounced oscillations is noticed with "hole burning" structure in the central peak due to nonzero phase of the initial amplitude (α).

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37

ŁAWRYNOWICZ, JULIAN, and AGNIESZKA NIEMCZYNOWICZ. "LATTICE DYNAMICS IN RELATION TO CHAOS IN ZWANZIG-TYPE CHAINS." International Journal of Bifurcation and Chaos 23, no. 11 (November 2013): 1350183. http://dx.doi.org/10.1142/s0218127413501836.

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The Zwanzig's procedure (1960) for the description of the system of coupled harmonic oscillators is applied to the chain of interacting oscillations in order to find the adsorption power function, which is then determined by two terms: (i) the classical term proportional to the radio-frequency function squared and (ii) the additional term linear with respect to the radio-frequency magnetic field amplitude. From the physical point of view the first term is usually considered in oscillator effect. The second term found in the present paper via Zwanzig's procedure seems to be induced by fluctuations due to stochastic distributions of the oscillatory precession phases. It reflects well the chaos as described in a fractal approach of the first author (2012, paper joint with M. Nowak-Kȩpczyk and O. Suzuki).

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38

Singh, Ram Mehar. "Integrability of a Coupled Harmonic Oscillator in Extended Complex Phase Space." Interdisciplinary journal of Discontinuity, Nonlinearity and Complexity 4, no. 1 (March 2015): 35–48. http://dx.doi.org/10.5890/dnc.2015.03.004.

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39

Singh, Ram Mehar, S. B. Bhardwaj, Kushal Sharma, Richa Rani, and Fakir Chand. "Integrability of a time dependent coupled harmonic oscillator in higher dimensions." Interdisciplinary journal of Discontinuity, Nonlinearity and Complexity 7, no. 1 (March 2018): 81–94. http://dx.doi.org/10.5890/dnc.2018.03.007.

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40

Chiorescu, I., P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij. "Coherent dynamics of a flux qubit coupled to a harmonic oscillator." Nature 431, no. 7005 (September 2004): 159–62. http://dx.doi.org/10.1038/nature02831.

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41

Kuznetsov, M. "Coupled wave analysis of multiple waveguide systems: The discrete harmonic oscillator." IEEE Journal of Quantum Electronics 21, no. 12 (December 1985): 1893–98. http://dx.doi.org/10.1109/jqe.1985.1072607.

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42

Kao, Jhih-Yuan, and Chung-Hsien Chou. "Quantum entanglement in coupled harmonic oscillator systems: from micro to macro." New Journal of Physics 18, no. 7 (July 1, 2016): 073001. http://dx.doi.org/10.1088/1367-2630/18/7/073001.

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43

Chetouani, L., L. Guechi, T. F. Hammann, and M. Letlout. "Path integral for the damped harmonic oscillator coupled to its dual." Journal of Mathematical Physics 35, no. 3 (March 1994): 1185–91. http://dx.doi.org/10.1063/1.530634.

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44

Zúñiga, José, Adolfo Bastida, and Alberto Requena. "Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates." Journal of Physics B: Atomic, Molecular and Optical Physics 53, no. 2 (December 23, 2019): 025101. http://dx.doi.org/10.1088/1361-6455/ab56d1.

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45

Sasihithlu, Karthik. "Coupled harmonic oscillator model to describe surface-mode mediated heat transfer." Journal of Photonics for Energy 9, no. 03 (December 27, 2018): 1. http://dx.doi.org/10.1117/1.jpe.9.032709.

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46

Chew, Lock Yue, and Ning Ning Chung. "Quantum entanglement and squeezing in coupled harmonic and anharmonic oscillator systems." Journal of Russian Laser Research 32, no. 4 (July 2011): 331–37. http://dx.doi.org/10.1007/s10946-011-9221-3.

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47

Hacker, Nir, and Boris A. Malomed. "Nonlinear Dynamics of Wave Packets in Tunnel-Coupled Harmonic-Oscillator Traps." Symmetry 13, no. 3 (February 25, 2021): 372. http://dx.doi.org/10.3390/sym13030372.

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We consider a two-component linearly coupled system with the intrinsic cubic nonlinearity and the harmonic-oscillator (HO) confining potential. The system models binary settings in BEC and optics. In the symmetric system, with the HO trap acting in both components, we consider Josephson oscillations (JO) initiated by an input in the form of the HO’s ground state (GS) or dipole mode (DM), placed in one component. With the increase of the strength of the self-focusing nonlinearity, spontaneous symmetry breaking (SSB) between the components takes place in the dynamical JO state. Under still stronger nonlinearity, the regular JO initiated by the GS input carries over into a chaotic dynamical state. For the DM input, the chaotization happens at smaller powers than for the GS, which is followed by SSB at a slightly stronger nonlinearity. In the system with the defocusing nonlinearity, SSB does not take place, and dynamical chaos occurs in a small area of the parameter space. In the asymmetric half-trapped system, with the HO potential applied to a single component, we first focus on the spectrum of confined binary modes in the linearized system. The spectrum is found analytically in the limits of weak and strong inter-component coupling, and numerically in the general case. Under the action of the coupling, the existence region of the confined modes shrinks for GSs and expands for DMs. In the full nonlinear system, the existence region for confined modes is identified in the numerical form. They are constructed too by means of the Thomas–Fermi approximation, in the case of the defocusing nonlinearity. Lastly, particular (non-generic) exact analytical solutions for confined modes, including vortices, in one- and two-dimensional asymmetric linearized systems are found. They represent bound states in the continuum.

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48

Lim, S. C., Chai Hok Eab, K. H. Mak, Ming Li, and S. Y. Chen. "Solving Linear Coupled Fractional Differential Equations by Direct Operational Method and Some Applications." Mathematical Problems in Engineering 2012 (2012): 1–28. http://dx.doi.org/10.1155/2012/653939.

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A new direct operational inversion method is introduced for solving coupled linear systems of ordinary fractional differential equations. The solutions so-obtained can be expressed explicitly in terms of multivariate Mittag-Leffler functions. In the case where the multiorders are multiples of a common real positive number, the solutions can be reduced to linear combinations of Mittag-Leffler functions of a single variable. The solutions can be shown to be asymptotically oscillatory under certain conditions. This technique is illustrated in detail by two concrete examples, namely, the coupled harmonic oscillator and the fractional Wien bridge circuit. Stability conditions and simulations of the corresponding solutions are given.

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49

Georgiou, I. T., and I. B. Schwartz. "Slaving the In-Plane Motions of a Nonlinear Plate to Its Flexural Motions: An Invariant Manifold Approach." Journal of Applied Mechanics 64, no. 1 (March 1, 1997): 175–82. http://dx.doi.org/10.1115/1.2787270.

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We show that the in-plane motions of a nonlinear isotropic plate can be decoupled from its transverse motions. We demonstrate this decoupling by showing analytically and numerically the existence of a global nonlinear invariant manifold in the phase space of three nonlinearly coupled fundamental oscillators describing the amplitudes of the coupled fundamental modes. The invariant manifold carries a continuum of slow periodic motions. In particular, for any motion on the slow invariant manifold, the transverse oscillator executes a periodic motion and it slaves the in-plane oscillators into periodic motions of half its period. Furthermore, as the energy level of a motion on the slow manifold increases, the frequency of the largest harmonic of the in-plane motion approaches the in-plane natural frequencies.

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Nicu, Valentin Paul. "Revisiting an old concept: the coupled oscillator model for VCD. Part 1: the generalised coupled oscillator mechanism and its intrinsic connection to the strength of VCD signals." Physical Chemistry Chemical Physics 18, no. 31 (2016): 21202–12. http://dx.doi.org/10.1039/c6cp01282e.

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Abstract:

This work reports the development of a generalised coupled oscillator expression for VCD that is exact within the harmonic approximation and is applicable to all types of normal modes, regardless whether the considered molecule is symmetric or asymmetric.

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