Asymptotic Behaviour and Hopf Bifurcation of a Three-Dimensional Nonlinear Autonomous System

2002 ◽  
Vol 9 (2) ◽  
pp. 207-226
Author(s):  
Lenka Baráková
Keyword(s):  
Hopf Bifurcation ◽  
Autonomous System ◽  
Three Dimensional ◽  
Invariant Set ◽  
Concrete Type ◽  
Cubic System ◽  
Dynamic Version ◽  
Globally Attractive ◽  
Nonlinear Autonomous System ◽  
Positively Invariant Set

Abstract A three-dimensional real nonlinear autonomous system of a concrete type is studied. The Hopf bifurcation is analyzed and the existence of a limit cycle is proved. A positively invariant set, which is globally attractive, is found using a suitable Lyapunov-like function. Corollaries for a cubic system are presented. Also, a two-dimensional nonlinear system is studied as a restricted system. An application in economics to the Kodera's model of inflation is presented. In some sense, the model of inflation is an extension of the dynamic version of the neo-keynesian macroeconomic IS-LM model and the presented results correspond to the results for the IS-LM model.

scholarly journals Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four-Dimensional Hyperchaotic Financial System

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. C. Wei ◽  
J. F. Wang ◽  
A. Akgul
Keyword(s):  
Hopf Bifurcation ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Financial System ◽  
Invariant Set ◽  
Phase Portraits ◽  
Lyapunov Coefficient ◽  
Stable Periodic ◽  
Analytical Proof ◽  
Positively Invariant Set

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

Hidden Hyperchaotic Attractors in a Modified Lorenz–Stenflo System with Only One Stable Equilibrium

2014 ◽  
Vol 24 (10) ◽  
pp. 1450127 ◽  
Author(s):  
Zhouchao Wei ◽  
Wei Zhang
Keyword(s):  
Autonomous System ◽  
Stable Equilibrium ◽  
Initial Conditions ◽  
Invariant Set ◽  
Period Orbit ◽  
Quadratic Nonlinearities ◽  
Positively Invariant Set ◽  
Parameter Values ◽  
Ultimate Bound ◽  
New System

This paper reports the finding of a four-dimensional (4D) non-Sil'nikov autonomous system with three quadratic nonlinearities, which exhibits some behavior previously unobserved: hidden hyperchaotic attractors with only one stable equilibrium. The algebraical form of the non-Sil'nikov chaotic attractor is very similar to the hyperchaotic Lorenz–Stenflo system but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact this system has only one stable equilibrium, but can exhibit hidden hyperchaos, chaos, periodic orbit. Moreover, the coexistence of attracting sets can be obtained in the system for some parameter values and different initial conditions, such as hyperchaotic attractor and point, hyperchaotic attractor and period orbit. To further analyze the new system, the ultimate bound and positively invariant set for the modified hyperchaotic Lorenz–Stenflo system are also obtained. Moreover, the complete mathematical characterizations for 4D Hopf bifurcation are rigorously derived and studied.

scholarly journals A Mathematical Model Related To Chemostat With Variable Yields

2012 ◽  
Vol 60 (2) ◽  
pp. 147-152
Author(s):  
S. M. Sohel Rana ◽  
Ummi Kulsum
Keyword(s):  
Mathematical Model ◽  
Steady State ◽  
Hopf Bifurcation ◽  
Qualitative Analysis ◽  
Global Stability ◽  
Three Dimensional ◽  
Invariant Set ◽  
Chemostat Model ◽  
Local And Global Stability ◽  
Positive Invariant Set

In this paper, a three dimensional chemostat model with variable yields is studied. The properties of the steady state points, the local and global stability, the Hopf bifurcation and the positive invariant set for the system are investigated by qualitative analysis of differential equations.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11482 Dhaka Univ. J. Sci. 60(2): 147-152, 2012 (July)

An application of the theory of monotone systems to an electrical circuit

2002 ◽  
Vol 132 (3) ◽  
pp. 711-728 ◽  
Author(s):  
Luis A. Sánchez
Keyword(s):  
Systems Theory ◽  
Autonomous System ◽  
Three Dimensional ◽  
Electrical Circuit ◽  
Monotone Systems ◽  
Periodic Oscillations ◽  
Stable Periodic ◽  
Nonlinear Autonomous System

This paper considers the dynamics of a three-dimensional nonlinear autonomous system that models the behaviour of an electrical circuit. Results on the existence of stable periodic oscillations and the behaviour of Poincaré–Bendixon types are obtained. The work is based on a variation of classical monotone systems theory.

On the Analysis of Hopf Bifurcation Associated with a Two-Fold Zero Eigenvalue, Part 1: Autonomous System

1999 ◽  
Vol 121 (1) ◽  
pp. 101-104 ◽  
Author(s):  
M. Moh’d ◽  
K. Huseyin
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Autonomous System ◽  
Periodic Motions ◽  
Zero Eigenvalue ◽  
The Family ◽  
Dynamic Bifurcations ◽  
Formula Type

The static and dynamic bifurcations of an autonomous system associated with a twofold zero eigenvalue (of index one) are studied. Attention is focused on Hopf bifurcation solutions in the neighborhood of such a singularity. The family of limit cycles are analyzed fully by applying the formula type results of the Intrinsic Harmonic Balancing method. Thus, parameter-amplitude and amplitude-frequency relationships as well as an ordered form of approximations for the periodic motions are obtained explicitly. A verification technique, with the aid of MAPLE, is used to show the consistency of ordered approximations.

On the Analysis of Hopf Bifurcation Associated with a Two-Fold Zero Eigenvalue, Part 2: Nonautonomous System

1999 ◽  
Vol 121 (1) ◽  
pp. 105-109 ◽  
Author(s):  
M. Moh’d ◽  
K. Huseyin
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Autonomous System ◽  
Critical Value ◽  
Harmonic Excitation ◽  
Nonautonomous System ◽  
External Excitation ◽  
Zero Eigenvalue ◽  
Bifurcation And Stability

This paper extends the bifurcation and stability analysis of the autonomous system considered in Part 1 to the case of a corresponding nonautonomous system. The effect of an external harmonic excitation on the Hopf bifurcation is studied via a modified Intrinsic Harmonic Balancing technique. It is observed that a shift in the critical value of the parameter occurs due to the external excitation. The analysis is carried out with the aid of MAPLE which is also instrumental in verifying the consistency of the approximations conveniently.

Three-Dimensional Hopf Bifurcation for a Class of Cubic Kolmogorov Model

2014 ◽  
Vol 24 (03) ◽  
pp. 1450036 ◽  
Author(s):  
Chaoxiong Du ◽  
Qinlong Wang ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Three Dimensional ◽  
Singular Values ◽  
Bifurcation Problem ◽  
Disturbed Condition ◽  
Differential Systems ◽  
Kolmogorov Model ◽  
Polynomial Differential Systems ◽  
Fine Focus

We study the Hopf bifurcation for a class of three-dimensional cubic Kolmogorov model by making use of our method (i.e. singular values method). We show that the positive singular point (1, 1, 1) of an investigated model can become a fine focus of 5 order, and moreover, it can bifurcate five small limit cycles under certain coefficients with disturbed condition. In terms of three-dimensional cubic Kolmogorov model, published references can hardly be seen, and our results are new. At the same time, it is worth pointing out that our method is valid to study the Hopf bifurcation problem for other three-dimensional polynomial differential systems.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

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