scholarly journals A survey on the blow-up method for fast-slow systems

2021 ◽  
pp. 115-160
Author(s):  
Hildeberto Jardón-Kojakhmetov ◽  
Christian Kuehn
Keyword(s):  
Algebraic Geometry ◽  
Blow Up ◽  
Geometric Theory ◽  
Geometric Approach ◽  
Van Der Pol Oscillator ◽  
Compact Form ◽  
Van Der Pol ◽  
Wide Range ◽  
Geometric Technique ◽  
Hyperbolic Points

In this document we review a geometric technique, called the blow-up method, as it has been used to analyze and understand the dynamics of fast-slow systems around non-hyperbolic points. The blow-up method, having its origins in algebraic geometry, was introduced to the study of fast-slow systems in the seminal work by Dumortier and Roussarie in 1996, whose aim was to give a geometric approach and interpretation of canards in the van der Pol oscillator. Following Dumortier and Roussarie, many efforts have been performed to expand the capabilities of the method and to use it in a wide range of scenarios. Our goal is to present in a concise and compact form those results that, based on the blow-up method, are now the foundation of the geometric theory of fast-slow systems with non-hyperbolic singularities. Due to their great importance in the theory of fast-slow systems, we cover fold points as one of the main topics. Furthermore, we also present several other singularities such as Hopf, pitchfork, transcritical, cusp, and Bogdanov-Takens, in which the blow-up method has been proved to be extremely useful. Finally, we survey further directions as well as examples of specific applied models, where the blow-up method has been used successfully.

Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude

2015 ◽  
Vol 24 (24) ◽  
pp. 5726-5734 ◽  
Author(s):  
Mustapha Hamdi ◽  
Mohamed Belhaq
Keyword(s):  
Resonant Frequency ◽  
Parametric Resonance ◽  
Large Range ◽  
Excitation Frequency ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Analytical Approximations ◽  
Wide Range ◽  
Periodic Delay ◽  
Time Periodic

Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude is investigated in this paper. The case where the delay amplitude in the position is modulated with a certain amplitude and a resonant frequency is considered. Application of the double perturbation method enables approximation of the amplitudes of quasi-periodic vibrations of the oscillator near a parametric resonance for which the frequency of the delay amplitude modulation is near twice the natural frequency of the oscillator. Analytical approximations supported by numerical simulations provide the regions in parameter space where quasi-periodic vibrations exist. Results show that the modulation of the delay amplitude in the position not only gives birth to quasi-periodic vibrations in a large range of parameters, but also with broadband large-amplitude covering a wide range of excitation frequency.

scholarly journals Controlling Canard Cycles

2021 ◽  
Author(s):  
Hildeberto Jardón-Kojakhmetov ◽  
Christian Kuehn
Keyword(s):  
Mixed Mode ◽  
Blow Up ◽  
Van Der Pol Oscillator ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Van Der Pol ◽  
Critical Manifold ◽  
Mixed Mode Oscillations ◽  
Special Solutions ◽  
And Control

AbstractCanard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator.

NUMERICAL SOLUTION FOR DUFFING-VAN DER POL OSCILLATOR VIA BLOCK METHOD

2020 ◽  
Vol 10 (1) ◽  
pp. 1857-8365
Author(s):  
A. F. Nurullah ◽  
M. Hassan ◽  
T. J. Wong ◽  
L. F. Koo
Keyword(s):  
Numerical Solution ◽  
Van Der Pol Oscillator ◽  
Block Method ◽  
Van Der Pol

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (02) ◽  
pp. 399-404 ◽  
Author(s):  
T. SÜNNER ◽  
H. SAUERMANN
Keyword(s):  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Dynamical Behaviour ◽  
Van Der Pol Oscillator ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

Sign in / Sign up

Export Citation Format

Share Document