Regularization and singular perturbation techniques for non-smooth systems

Teixeira, Marco Antônio; Silva, Paulo Ricardo da

doi:10.1016/j.physd.2011.06.022

Cited by 68 publications

(61 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The best understood of these is regularization by smoothing, in which the discontinuity is replaced by a steep sigmoid transition, and for which basic results exist describing how such systems approximate discontinuous systems [25,30]. Therefore when simulating examples of exit point behaviour for illustrative purposes only, we shall use smoothed out approximations of the discontinuous vector field as described in the text, and let the numerical integrator choose the path through the intersection as a numerical experiment.…”

Section: (Iv) (Iii)mentioning

confidence: 99%

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

Jeffrey

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

The collapse of flows onto hypersurfaces where their vector fields are discontinuous creates highly robust states called sliding modes. The way flows exit from such sliding modes can lead to complex and interesting behaviour about which little is currently known. Here we examine the basic mechanisms by which a flow exits from sliding, either along a switching surface, or along the intersection of two switching surfaces, with a view to understanding sliding and exit when many switches are involved. On a single switching surface, exit occurs via tangency of the flow to the switching surface. Along an intersection of switches, exit can occur at a tangency with a lower codimension sliding flow, or by a spiralling of the flow that exhibits geometric divergence (infinite steps in finite time). Determinacy-breaking can occur where a singularity creates a set-valued flow in an otherwise deterministic system, and we resolve such dynamics as far as possible by blowing up the switching surface into a switching layer. We show preliminary simulations exploring the role of determinacy-breaking events as organizing centres of local and global dynamics.Switching is found in dynamical models of wideranging applications, from mechanics and geophysics to biological growth and ecology. Switches occur between different dynamical laws whenever certain thresholds are encountered. In this paper we consider how systems behave when they exit from highly constrained states sliding along those thresholds or intersections thereof. For one or two switches we examine the basic mechanisms of exit. In particular we show that exit from sliding is not always deterministic, and we describe the main features of determinacy-breaking exit points. Example simulations that illustrate the theoretical results as novel dynamical phenomena are given.

show abstract

Section: (Iv) (Iii)mentioning

confidence: 99%

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

Jeffrey

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

show abstract

“…With 0 < q 2 < q 1 < 1, we emphasize the reduced benefit that the predator obtains from the alternative prey compared to its preferred prey. The difference in the benefit is also where the assumed tradeoff lies, as the alternative multiple time scales by "blowing up" the switching boundary [41,44]. In this work, we do not consider regularizations that include multiple time scales.…”

Section: Methodsmentioning

confidence: 99%

Inferring parameters of prey switching in a 1 predator–2 prey plankton system with a linear preference tradeoff

Piltz

Harhanen

Porter

et al. 2018

Journal of Theoretical Biology

View full text Add to dashboard Cite

We construct two ordinary-differential-equation models of a predator feeding adaptively on two prey types, and we evaluate the models' ability to fit data on freshwater plankton. We model the predator's switch from one prey to the other in two different ways: (i) smooth switching using a hyperbolic tangent function; and (ii) by incorporating a parameter that changes abruptly across the switching boundary as a system variable that is coupled to the population dynamics. We conduct linear stability analyses, use approximate Bayesian computation (ABC) combined with a population Monte Carlo (PMC) method to fit model parameters, and compare model results quantitatively to data for ciliate predators and their two algal prey groups collected from Lake Constance on the German-Swiss-Austrian border. We show that the two models fit the data well when the smooth transition is steep, supporting the simplifying assumption of a discontinuous prey-switching behavior for this scenario. We thus conclude that prey switching is a possible mechanistic explanation for the observed ciliate-algae dynamics in Lake Constance in spring, but that these data cannot distinguish between the details of prey switching that are encoded in these different models.

show abstract

“…in [17,20,22,27]. To-date these typically follow the Sotomayor-Teixeira [25] approach, and restrict the class of smoothed systems studied to those with strictly linear dependence on the switching parameter (i.e.…”

Section: In Closingmentioning

confidence: 99%

Hidden Bifurcations and Attractors in Nonsmooth Dynamical Systems

Jeffrey

2016

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

We investigate the role of hidden terms at switching surfaces in piecewise smooth vector fields. Hidden terms are zero everywhere except at the switching surfaces, but appear when blowing up the switching surface into a switching layer. When discontinuous systems do surprising things, we can often make sense of them by extending our intuition for smooth system to the switching layer. We illustrate the principle here with a few attractors that are hidden inside the switching layer, being evident in the flow, despite not being directly evident in the vector field outside the switching surface. These can occur either at a single switch (where we will introduce hidden terms somewhat artificially to demonstrate the principle), or at the intersection of multiple switches (where hidden terms arise inescapably). A more subtle role of hidden terms is in bifurcations, and we revisit some simple cases from previous literature here, showing that they exhibit degeneracies inside the switching layer, and that the degeneracies can be broken using hidden terms. We illustrate the principle in systems with one or two switches.

show abstract

Regularization and singular perturbation techniques for non-smooth systems

Cited by 68 publications

References 12 publications

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking

Inferring parameters of prey switching in a 1 predator–2 prey plankton system with a linear preference tradeoff

Hidden Bifurcations and Attractors in Nonsmooth Dynamical Systems

Contact Info

Product

Resources

About