Analysis on an Oscillator With Delayed Feedback Near Double Hopf Bifurcations

2001 ◽  
Author(s):  
P. Yu ◽  
Y. Yuan
Keyword(s):  
Time Delay ◽  
Nonlinear Oscillator ◽  
Analytical Approach ◽  
Complex Dynamics ◽  
Integration Scheme ◽  
Hopf Bifurcations ◽  
External Forcing ◽  
Chaotic Motions ◽  
System Parameters ◽  
Numerical Integration Scheme

Abstract This paper considers the effect of time delayed feedbacks in a nonlinear oscillator with external forcing. The particular attention is focused on the case where the corresponding linear system has two pairs of purely imaginary eigenvalues at a critical point, leading to double Hopf bifurcations. An analytical approach is used to find the explicit expressions for the critical values of the system parameters at which non-resonant or resonant Hopf bifurcations may occur. A fourth-order Ronge-Kutta numerical integration scheme is applied to obtain the dynamical solutions in the vicinity of the critical points. Both the cases with and without the external forcing are considered. It has been found the system exhibits very rich complex dynamics including periodic, quasi-periodic and chaotic motions. Moreover, a sensitivity analysis is carried out to show that chaotic motions are very sensitive to the time delay. This suggests that the time delay can be used: (1) to control bifurcations and chaos; and (2) to generate bifurcations and chaos.

TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY

2012 ◽  
Vol 22 (03) ◽  
pp. 1250060 ◽  
Author(s):  
J. C. JI ◽  
X. Y. LI ◽  
Z. LUO ◽  
N. ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Critical Time ◽  
Zero Solution ◽  
Delayed Feedback Control ◽  
Hopf Bifurcations ◽  
Centre Manifold ◽  
Normal Form Theory

The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf–Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf–Hopf interactions.

Bifurcation Behavior of a Supercavitating Vehicle

2006 ◽  
Author(s):  
Guojian Lin ◽  
Balakumar Balachandran ◽  
Eyad H. Abed
Keyword(s):  
Complex Dynamics ◽  
Numerical Study ◽  
Period Doubling ◽  
Linear Feedback ◽  
Hopf Bifurcations ◽  
Linear Feedback Control ◽  
Chaotic Motions ◽  
Supercavitating Vehicle ◽  
Bifurcation Behavior ◽  
Period Doubling Bifurcations

In this effort, a numerical study of the bifurcation behavior of a supercavitating vehicle is conducted. The nonsmoothness of this system is due to the planing force acting on the vehicle. With a focus on dive-plane dynamics, bifurcations with respect to a quasi-static variation of the cavitation number are studied. The system is found to exhibit rich and complex dynamics including nonsmooth bifurcations such as the grazing bifurcation and smooth bifurcations such as Hopf bifurcations, cyclic-fold bifurcations, and period-doubling bifurcations, chaotic attractors, transient chaotic motions, and crises. The tailslap phenomenon of the supercavitating vehicle is identified as a consequence of the Hopf bifurcation followed by a grazing event. It is shown that the occurrence of these bifurcations can be delayed or triggered earlier by using dynamic linear feedback control aided by washout filters.

WEAK RESONANT DOUBLE HOPF BIFURCATIONS IN AN INERTIAL FOUR-NEURON MODEL WITH TIME DELAY

2012 ◽  
Vol 22 (01) ◽  
pp. 63-75 ◽  
Author(s):  
JUHONG GE ◽  
JIAN XU
Keyword(s):  
Time Delay ◽  
Neuron Model ◽  
Hopf Bifurcations ◽  
Bidirectional Associative Memory ◽  
System Parameters ◽  
Dynamical Behaviors ◽  
Delayed System ◽  
Incremental Scheme ◽  
The Relationship ◽  
Theoretical Results

In this paper, a four-neuron delayed bidirectional associative memory (BAM) model with inertia is considered. Weak resonant double Hopf bifurcations are completely analyzed in the parameter space of the coupling weight and the coupling delay by the perturbation-incremental scheme (PIS). Numerical simulations are given for justifying the theoretical results. To the best of our knowledge, the paper is the first one to introduce inertia to a four-neuron delayed system and clarify the relationship between system parameters and dynamical behaviors.

A Chaotic Time-Delay System with Saturation Nonlinearity

2017 ◽  
Vol 6 (3) ◽  
pp. 111-129 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sundarapandian Vaidyanathan
Keyword(s):  
Time Delay ◽  
Complex Dynamics ◽  
Chaotic Signal ◽  
Delay System ◽  
Delay Systems ◽  
Maximum Lyapunov Exponent ◽  
Time Delay Systems ◽  
Time Delay System ◽  
System Parameters ◽  
Saturation Nonlinearity

Complex dynamics are observed in time-delay systems because the presence of time delay could induce unexpected oscillations. Therefore, time-delay systems are effective for constructing chaotic signal generators which have used in various engineering applications. In this paper, a new system with a single scalar time delay and a saturation nonlinearity is introduced. Dynamics of such time-delay system are investigated by using phase planes, bifurcation diagrams and the maximum Lyapunov exponent with the variance of system parameters. It is interesting that the time-delay system can generate double-scroll chaotic attractors despite its elegant model. Circuitry of the system is also presented to show the feasibility of the theoretical model.

A MODIFIED GENERALIZED LORENZ-TYPE SYSTEM AND ITS CANONICAL FORM

2009 ◽  
Vol 19 (06) ◽  
pp. 1931-1949 ◽  
Author(s):  
QIGUI YANG ◽  
KANGMING ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Special Form ◽  
Topological Structure ◽  
Complex Dynamics ◽  
Type System ◽  
Chaotic Attractors ◽  
Hopf Bifurcations ◽  
Compound Structure ◽  
Two Parameters ◽  
First Time ◽  
New System

In this paper, a modified generalized Lorenz-type system is introduced, which is state-equivalent to a simple and special form, and is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, two classes of new chaotic attractors are found for the first time in the literature, which are similar but nonequivalent in topological structure. To further understand the complex dynamics of the new system, some basic properties such as Lyapunov exponents, Hopf bifurcations and compound structure of the attractors are analyzed and demonstrated with careful numerical simulations.

scholarly journals Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Feras K. Alfosail ◽  
Amal Z. Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Chaotic Motions ◽  
Different Types ◽  
Quadratic Nonlinearities ◽  
Multiple Scales Perturbation Method ◽  
Good Agreement ◽  
Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

Sequences of Global Bifurcations and the Related Outcomes after Crisis of the Resonant Attractor in a Nonlinear Oscillator

1997 ◽  
Vol 07 (11) ◽  
pp. 2437-2457 ◽  
Author(s):  
W. Szemplińska-Stupnicka ◽  
E. Tyrkiel
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Resonance ◽  
Basins Of Attraction ◽  
Global Bifurcations ◽  
System Behavior ◽  
Coexisting Attractors ◽  
System Parameters

The problem of the system behavior after annihilation of the resonant attractor in the region of the nonlinear resonance hysteresis is considered. The sequences of global bifurcations, in connection with the associated metamorphoses of basins of attraction of coexisting attractors, are examined. The study allows one to reveal the mechanism that governs the phenomenon of the post crisis ensuing transient trajectory to settle onto one or another remote attractor. The problem is studied in detail for the twin-well potential Duffing oscillator. The boundary which splits the considered region of system parameters into two subdomains, where the outcome is unique or the two outcomes are possible, is defined.

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