SUBCRITICAL HOPF BIFURCATION AT PHONATION ONSET

Abstract In this paper, an efficient numerical method consisting of the real mode component mode synthesis (CMS) model reduction, shooting method with parallel computing, and Floquet analysis was developed for nonlinear rotordynamics analysis of a flexible rotor supported by a 4-lobe flexure pivot tilting pad journal bearing (FPTPJB) in load-on-pad (LOP) and load-between-pad (LBP) orientations in comparison to a fixed profile journal bearing (JB) of the same pad geometry. The method used the rotor's finite elements and bearing forces obtained from directly solving the Reynolds equation to determine the limit cycles and Hopf bifurcation types. For the investigated rotor and bearing parameters, the numerical results indicated that the onset speed of instability (OSI) of FPTPJB is considerably higher than that of JB of the same orientation. Also, FPTPJB in LOP orientation yielded higher OSI than the LBP one, whereas the OSI of JB in LOP orientation was substantially higher than the LBP counterpart. Nonlinear calculation results indicated that all bearing types and orientations gave subcritical Hopf bifurcation. The FPTPJB in LOP orientation produced the largest stable operating region, whereas the JB in LBP configuration yield the smallest one. The experiment showed subcritical Hopf bifurcation occurred at speed close to the calculated OSI in all cases except FPTPJB in LOP orientation that the OSI is higher than the maximum test rig speed. The whirling orbit had the same frequency as the first critical speed and precessed in the direction of shaft rotation.

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Modelling HIV dynamics with cell-to-cell transmission and CTL response

10.22541/au.163252606.62193511/v1 ◽

2021 ◽

Author(s):

Zirui Zhu ◽

Ranchao Wu ◽

Yu Yang ◽

Yancong Xu

Keyword(s):

Hopf Bifurcation ◽

Cell Interaction ◽

Backward Bifurcation ◽

Hiv Treatment ◽

Hiv Model ◽

Ctl Response ◽

Hiv Dynamics ◽

Subcritical Hopf Bifurcation ◽

Cell Transmission ◽

New Type

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell-to-cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so-called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov-Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell-to-cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so-called new type bifurcation persist. Finally, some conclusions and discussions are also given.

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Rendering a subcritical Hopf bifurcation supercritical

Journal of Fluid Mechanics ◽

10.1017/s0022112096000675 ◽

1996 ◽

Vol 317 ◽

pp. 91-109 ◽

Cited By ~ 44

Author(s):

Po Ki Yuen ◽

Haim H. Bau

Keyword(s):

Hopf Bifurcation ◽

Limit Cycle ◽

Thermal Convection ◽

Control Strategy ◽

Chaotic Motion ◽

Feedback Controller ◽

Nonlinear Feedback ◽

Naturally Occurring ◽

Subcritical Hopf Bifurcation ◽

Stable Periodic

It is demonstrated experimentally and theoretically that through the use of a nonlinear feedback controller, one can render a subcritical Hopf bifurcation supercritical and thus dramatically modify the nature of the flow in a thermal convection loop heated from below and cooled from above. In particular, we show that the controller can replace the naturally occurring chaotic motion with a stable, periodic limit cycle. The control strategy consists of sensing the deviation of fluid temperatures from desired values at a number of locations inside the loop and then altering the wall heating to counteract such deviations.

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Study of Dynamic Characteristics of Friction: A Comparative Analysis of the Velocity Dependent and the LuGre Friction Model

Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2005-84533 ◽

2005 ◽

Author(s):

Firoz Ali Jafri ◽

David F. Thompson

Keyword(s):

Hopf Bifurcation ◽

Friction Model ◽

Continuation Methods ◽

Bifurcation Parameter ◽

Lugre Model ◽

Lugre Friction Model ◽

Subcritical Hopf Bifurcation ◽

Fold Bifurcation ◽

Dependent Model ◽

Lugre Friction

In this paper, we conduct numerical analysis to study the effects of friction on the dynamic response of a single degree of freedom mechanical system. Two different friction models, the velocity dependent friction model and the LuGre friction model, have been used to model the friction interface. Bifurcation analysis has been conducted using equilibrium and limit cycle continuation methods. With system viscous damping as the bifurcation parameter, a reverse subcritical Hopf bifurcation is observed in the case of velocity dependent model. In the case of the LuGre model for the same bifurcation parameter, a reverse supercritical Hopf bifurcation is observed at lower velocities but at higher velocities it changes to a reverse subcritical Hopf bifurcation. A fold bifurcation of the limit cycles is also seen at higher velocities for the LuGre model.

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Subcritical bifurcation and bistability in thermoacoustic systems

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.514 ◽

2013 ◽

Vol 715 ◽

pp. 210-238 ◽

Cited By ~ 32

Author(s):

Priya Subramanian ◽

R. I. Sujith ◽

P. Wahi

Keyword(s):

Hopf Bifurcation ◽

Linear Stability ◽

Multiple Scales ◽

Landau Equation ◽

Linear Instability ◽

Numerical Continuation ◽

Rijke Tube ◽

Subcritical Hopf Bifurcation ◽

Fold Bifurcation ◽

Parameter Values

AbstractThis paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.

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NEURONS ARE POISED NEAR THE EDGE OF CHAOS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412500988 ◽

2012 ◽

Vol 22 (04) ◽

pp. 1250098 ◽

Cited By ~ 70

Author(s):

LEON CHUA ◽

VALERY SBITNEV ◽

HYONGSUK KIM

Keyword(s):

Hopf Bifurcation ◽

Bifurcation Point ◽

Nonlinear Differential Equations ◽

Complex Conjugate ◽

Excitation Current ◽

Hopf Bifurcation Point ◽

Synaptic Excitation ◽

Edge Of Chaos ◽

Subcritical Hopf Bifurcation ◽

Dc Current

This paper shows the action potential (spikes) generated from the Hodgkin–Huxley equations emerges near the edge of chaos consisting of a tiny subset of the locally active regime of the HH equations. The main result proves that the eigenvalues of the 4 × 4 Jacobian matrix associated with the mathematically intractable system of four nonlinear differential equations are identical to the zeros of a scalar complexity function from complexity theory. Moreover, we show the loci of a pair of complex-conjugate zeros migrate continuously as a function of an externally applied DC current excitation emulating the net synaptic excitation current input to the neuron. In particular, the pair of complex-conjugate zeros move from a subcritical Hopf bifurcation point at low excitation current to a super-critical Hopf bifurcation point at high excitation current. The spikes are generated as the excitation current approaches the vicinity of the edge of chaos, which leads to the onset of the subcritical Hopf bifurcation regime. It follows from this in-depth qualitative analysis that local activity is the origin of spikes.

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Third-Order Memristive Morris–Lecar Model of Barnacle Muscle Fiber

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417300154 ◽

2017 ◽

Vol 27 (04) ◽

pp. 1730015 ◽

Cited By ~ 9

Author(s):

Vetriveeran Rajamani ◽

Maheshwar PD. Sah ◽

Zubaer Ibna Mannan ◽

Hyongsuk Kim ◽

Leon Chua

Keyword(s):

Hopf Bifurcation ◽

Muscle Fiber ◽

Limit Cycle Oscillation ◽

Potassium Ions ◽

Stable Equilibrium Point ◽

Third Order ◽

The Third ◽

Barnacle Muscle ◽

Subcritical Hopf Bifurcation ◽

Barnacle Muscle Fiber

This paper presents a detailed analysis of various oscillatory behaviors observed in relation to the calcium and potassium ions in the third-order Morris–Lecar model of giant barnacle muscle fiber. Since, both the calcium and potassium ions exhibit all of the characteristics of memristor fingerprints, we claim that the time-varying calcium and potassium ions in the third-order Morris–Lecar model are actually time-invariant calcium and potassium memristors in the third-order memristive Morris–Lecar model. We confirmed the existence of a small unstable limit cycle oscillation in both the second-order and the third-order Morris–Lecar model by numerically calculating the basin of attraction of the asymptotically stable equilibrium point associated with two subcritical Hopf bifurcation points. We also describe a comprehensive analysis of the generation of oscillations in third-order memristive Morris–Lecar model via small-signal circuit analysis and a subcritical Hopf bifurcation phenomenon.

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Chaotic Dynamics by Some Quadratic Jerk Systems

Axioms ◽

10.3390/axioms10030227 ◽

2021 ◽

Vol 10 (3) ◽

pp. 227

Author(s):

Mei Liu ◽

Bo Sang ◽

Ning Wang ◽

Irfan Ahmad

Keyword(s):

Hopf Bifurcation ◽

Cross Sections ◽

Chaotic Dynamics ◽

Stable Equilibrium ◽

Period Doubling ◽

Dynamical Evolution ◽

Chaotic Attractors ◽

Bifurcation Diagrams ◽

Hidden Attractors ◽

Subcritical Hopf Bifurcation

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

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