Analytical approximation of the canard explosion in a van der Pol system with the nonlinear time transformation method

Algaba, Antonio; Chung, Kwok Wai; Qin, Bo-Wei; Rodríguez-Luis, Alejandro J.

doi:10.1016/j.physd.2020.132384

Cited by 19 publications

(10 citation statements)

References 34 publications

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“…If System (11) has two pairs of purely imaginary eigenvalues Λ = {±iω 01 , ±iω 02 } and other eigenvalues are negative, the phase space C can be divided into two subspaces. That is, C = P Λ ⊕ Q Λ .…”

Section: Computation Of Normal Form and Center-manifold Reductionmentioning

confidence: 99%

“…We study the normal form of System (29), which is reduced by the central manifold of Equation (11). It can reflect some properties of System (11).…”

Section: Classification Of Dynamical Behavioursmentioning

confidence: 99%

“…Then they revealed that the nonlinear restoring forces in a pair of van der Pol oscillators can induce a transient chaotic route by a small disturbance to the amplitude of an oscillator. Algaba et al [11] investigated canard explosion in van der Pol electronic oscillators. They developed a new effective program which can calculate the approximate value of critical parameters of any desired order.…”

Section: Introductionmentioning

confidence: 99%

“…It can be seen that in the literature [8][9][10][11], we have studied the branching, chaos, canard explosion, and other phenomena of the coupled van der Pol oscillators without time delay from the phenomenology. However, they still lack the analysis of the theory behind the coupled van der Pol oscillators.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Chen¹,

Qian²

2021

Mathematics

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In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV.

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Section: Computation Of Normal Form and Center-manifold Reductionmentioning

confidence: 99%

“…We study the normal form of System (29), which is reduced by the central manifold of Equation (11). It can reflect some properties of System (11).…”

Section: Classification Of Dynamical Behavioursmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Chen¹,

Qian²

2021

Mathematics

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“…Because of the symmetry of the Van der Pol model, the calculated critical parameter (truncated at first order) a c (ǫ) = 1 − ǫ/8 for the canard implosion yields a c (0.01) = 0.99875, which is again in excellent agreement with the numerical value 0.998740... shown in Fig.4. Higher-order corrections to the Van der Pol canard parameter a c (ǫ) = 1 − ǫ/8 − 3 ǫ 2 /32 − 173 ǫ 3 /1024 − • • • can be computed up to arbitrary order[14] but they are not needed in what follows.…”

mentioning

confidence: 99%

Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions

Brizard

Berry

2021

Preprint

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The asymptotic limit-cycle analysis of mathematical models for oscillating chemical reactions is presented. In this work, after a brief presentation of mathematical preliminaries applied to the biased Van der Pol oscillator, we consider a two-dimensional model of the Chlorine dioxide Iodine Malonic-Acid (CIMA) reactions and the three-dimensional and two-dimensional Oregonator models of the Belousov-Zhabotinsky (BZ) reactions. Explicit analytical expressions are given for the relaxation-oscillation periods of these chemical reactions that are accurate within 5% of their numerical values. In the two-dimensional CIMA and Oregonator models, we also derive critical parameter values leading to canard explosions and implosions in their associated limit cycles.

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Oscillatory systems with two degrees of freedom and van der Pol coupling: Analytical approach

Kraljevic,

Zukovic,

Cveticanin

2024

Math Methods in App Sciences

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In this paper steady‐state vibrations of the two‐degrees‐of‐freedom oscillatory systems with van der Pol coupling are investigated. The model is a system of two differential equations with weak nonlinearity. A new solving procedure based on D′Alembert's method and the method of time‐variable amplitude and phase is developed. The main advantage of the method in comparison to others is that it gives the solution of the system of two coupled weak nonlinear equations in the form that is simple to analyze, as it has the same form as the solution of the corresponding system of linear equations. In the paper two types of systems are considered: one, a two‐mass system with two degrees of freedom, and second, the one‐mass system with two degrees of freedom. The torsional vibrations of a two‐mass system and vibrations of a Jeffcott rotor with two‐degrees‐of‐freedom are analyzed. Analytically obtained results are numerically tested. It is obtained that the difference between analytic and numeric results is small and almost negligible. As the accuracy of the analytic solution is high, it is suitable for application in technics and engineering. Conclusions about steady‐state self‐sustainable oscillators, orbital, and limit cycle motions are given.

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Analytical approximation of the canard explosion in a van der Pol system with the nonlinear time transformation method

Cited by 19 publications

References 34 publications

Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions

Oscillatory systems with two degrees of freedom and van der Pol coupling: Analytical approach

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