Bifurcation and degenerate decomposition in multiple time scale dynamical systems

Guckenheimer, John

doi:10.1887/0750308621/b1112c1

Cited by 10 publications

(23 citation statements)

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“…This orbit Norm of x canards start to form (17) canards jump back (16) no canards (18) no canards (10) maximal canard (15) canards jump across (14) no canard (19) appears to be very close to one with a tangency with the surfaces x = ±1. We conjecture that the relationship between a homoclinic bifurcation of the reduced system and the presence of saddle-node bifurcations in the "full" system is a general one [13]. We also note that the apparent vertex of the bifurcation curve at the point labeled (11) seems to be a smooth fold in a blowup of the region near this turning point in Figure 28.…”

Section: Comparisons Of the Reduced And Full Systems: Numerical Calcumentioning

confidence: 68%

“…The symbols are directly related to the a, b, n we used to describe the jumps of Norm of x canards start to form (11) canards jump back (12) no canards (18) no canards (10) maximal canard (13) canards jump across (14) no canard (19) maximal canard (15) canards jump back (16) canards start to form (17) see magnified view the periodic orbit in Figure 27. The subshift allows arbitrary sequences of these symbols with the single restriction that there is a limit on the length of consecutive n's.…”

Section: Comparisons Of the Reduced And Full Systems: Numerical Calcumentioning

confidence: 99%

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The Forced van der Pol Equation II: Canards in the Reduced System

Bold¹,

Edwards²,

Guckenheimer³

et al. 2003

SIAM J. Appl. Dyn. Syst.

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Abstract. This is the second in a series of papers about the dynamics of the forced van der Pol oscillator [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1-35].The first paper described the reduced system, a two dimensional flow with jumps that reflect fast trajectory segments in this vector field with two time scales. This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator. This extension of the reduced system serves as a template for approximating the full nonwandering set of the forced van der Pol oscillator for large sets of parameter values, including parameters for which the system is chaotic. We analyze some bifurcations in the extension of the reduced system, building upon our previous work in [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1-35]. We conclude with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduced system. These comparisons demonstrate numerically the validity of results we derive from the study of canards in the reduced system.

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Section: Comparisons Of the Reduced And Full Systems: Numerical Calcumentioning

confidence: 68%

Section: Comparisons Of the Reduced And Full Systems: Numerical Calcumentioning

confidence: 99%

The Forced van der Pol Equation II: Canards in the Reduced System

Bold¹,

Edwards²,

Guckenheimer³

et al. 2003

SIAM J. Appl. Dyn. Syst.

Self Cite

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“…Thus, the boundary layer system has an exponentially stable equilibrium point as long as and are constant. In the zero-epsilon approximation ( = 0), the zero-order approximation of the slow manifold, that is, critical manifold [14], is given by…”

Section: Algebraic Estimate Of Tonoplast Ordermentioning

confidence: 99%

Adaptive Estimation of Biological Rhythm in Crassulacean Acid Metabolism with Critical Manifold

Matsuo

Totoki

Suemitsu

2013

ISRN Applied Mathematics

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The mechanism of endogenous circadian photosynthesis oscillations of plants performing crassulacean acid metabolism (CAM) is investigated in terms of a nonlinear theoretical model. Blasius et al. used throughout continuous time differential equations which adequately reflect the CAM dynamics. The model shows regular endogenous limit cycle oscillations that are stable for a wide range of temperatures in a manner that complies well with experimental data. In this paper, we pay attention to the approximation of the fast modes of the CAM dynamics. Using the zero-epsilon approximation of the slow manifold, we derive the critical manifold that is defined by two algebraic nonlinear equations. The critical manifold allows us to give the algebraic estimate of the order of the tonoplast membrane. The dynamic equation of the order of the tonoplast membrane includes the nonlinear function that gives the equilibrium value of the lipid order of tonoplast functions as a hysteresis switch. We identify the nonlinear function with the measurement signals. Using the basis function expansion of the nonlinear and the critical manifold, we propose an adaptive observer to estimate the tonoplast order and the nonlinear function.

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“…See also Arnold et al [1] and Guckenheimer [7]. The local analysis of folded saddles describes the geometry of the flow of the "full" three dimensional system near the folded saddle.…”

Section: Folded Saddlesmentioning

confidence: 99%

Bifurcations of Relaxation Oscillations Near Folded Saddles

Hoffman

Weckesser

2005

Int. J. Bifurcation Chaos

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Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slow-fast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions that occur in generic one parameter families of relaxation oscillations. This paper investigates the bifurcations that are associated with one type of degeneracy that occurs in systems with two slow variables, namely orbits that become homoclinic to a folded saddle.

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Bifurcation and degenerate decomposition in multiple time scale dynamical systems

Cited by 10 publications

References 0 publications

The Forced van der Pol Equation II: Canards in the Reduced System

The Forced van der Pol Equation II: Canards in the Reduced System

Adaptive Estimation of Biological Rhythm in Crassulacean Acid Metabolism with Critical Manifold

Bifurcations of Relaxation Oscillations Near Folded Saddles

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