Characterization of bifurcations in the Zeeman catastrophe machine using singular value decomposition

Alvin Penner

Chaos, Solitons & Fractals, 2021, vol. 148, issue C

Abstract: Stable limit cycles of the dynamic Zeeman Catastrophe Machine are calculated and classified. As the geometry of the device is changed, they evolve into different types at discrete bifurcation points. The process is normally presented as a bifurcation diagram using a stroboscopic map, but this does not provide enough information to uniquely identify the bifurcation event. Three types of bifurcation are common: period-doubling, symmetry-breaking, and turning points. A method is developed to distinguish between these types by calculating the response to a change in device geometry. Assuming that the limit cycle has already been calculated, the response to a change in geometry is defined as the solution of a linear second-order differential equation. The equation is solved in finite-difference form by defining a matrix which contains the second-order response to a change in the limit cycle, as well as an augmented matrix which contains the response to a change in device geometry. At a bifurcation, the solution can be analyzed using singular value decomposition of these two matrices. The resulting singular values contain enough information to uniquely identify, and characterize, each type of event. The corresponding singular vectors can also be used to uniquely characterize the symmetry changes that occur in the response to change at each event.

Keywords: Zeeman catastrophe machine; chaos; Lyapunov exponent; limit cycle; symmetry-breaking; period-doubling; imperfect bifurcation; turning point; tangent space; singular value decomposition (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0960077921004082
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:148:y:2021:i:c:s0960077921004082

DOI: 10.1016/j.chaos.2021.111054

Access Statistics for this article

Chaos, Solitons & Fractals is currently edited by Stefano Boccaletti and Stelios Bekiros

More articles in Chaos, Solitons & Fractals from Elsevier
Bibliographic data for series maintained by Thayer, Thomas R. ().