3. Related mathematical and physical foundation issues
An interesting, related question is: Can non-oriented Möbius strips appear on the spatial limit cycles of three-dimensional differential autonomous systems?
The answer is yes. This situation occurs in the theoretical study of period-doubling bifurcation of the spatial limit cycles.
This period-doubling bifurcation usually occurs when a parameter λ (different from the independent time t) of the corresponding system changes continuously and exceeds a critical value λ0. When a period-doubling bifurcation occurs, each local line segment of the critical limit cycle splits into two new local curves after the bifurcation. These new local curves are integrated to form a new limit cycle after the bifurcation. The length and period of the new limit cycle are almost twice that before the bifurcation. Currently, if these two similar local curves are regarded as the two local boundaries of a strip, these local strips are integrated to form a closed Möbius strip.
At the beginning of the bifurcation, the band is just a critical limit cycle with no width. After the bifurcation, it has a width as the parameter λ0 changes. In practical studies, this parameter is usually artificially limited to a reasonable value, so the width of the band is fixed.
Naturally, the center of this belt can be regarded as the critical limit cycle at the beginning of the bifurcation.
Since this Möbius strip has only one boundary, which is just the bifurcated limit cycle, and which is still a closed curve, so this Möbius strip is non-directional!
For example, when a = 1, b = 0.4, the system (15) has a pair of limit cycles that are symmetric about the origin and have a rotation number equal to 1. The image of one of them is
Figure 35.
Numerical calculations show that when the value of parameter b is further reduced, period doubling occurs. When it starts to decrease, the Möbius strip is too narrow to be seen clearly. However, when a = 1, b = 0.39, the limit cycle of period doubling bifurcation is very clear, as shown in the figure below.
Figure 36.
If we combine Figures 35 and 36 with the period-doubling limit cycle diagrams for b = 0.392 and b = 0.393, we can show a perfectly non-directional Möbius strip:
(a) (b)
Figure 37.
Figure 37 shows the Möbius strip from two different viewing angles so that its non-directional nature can be more clearly shown.
The non-directional Möbius strip formed when the spatial limit cycle bifurcates given above provides an important analytical means for studying related theories.
We will now try to raise some new problems about differential topological invariants studied in differential geometry of three-dimensional space.
First, the closed curves discussed earlier in this article are not knotted, and there is no link between different closed curves. If these phenomena are considered together, it should be a topic that needs further research.
Secondly, space surfaces are much more complicated than space curves. This article does not discuss the details of surface theory but only points out that the theory is like curve theory, and there are intrinsic invariants like curvature and torsion, as well as similar basic theorems of surfaces. In particular, the Gaussian curvature K, as an intrinsic invariant of the surface, largely determines the basic properties of the surface, and forms Euclidean geometry and non-Euclidean geometry based on whether its value is always equal to zero, or a constant positive or negative number. Similar to curves, the author raises the following question from a physical perspective: There are often convex lumps or concave pits on smooth surfaces. Are the shape characteristics and quantity of these lumps or pits also differential topological invariants of the surface?
Another interesting question is related to particle physics. Relevant experts may know the concept of Dirac 's belt or the plate trick. As a three-dimensional topological object, the Dirac belt has played a certain positive role in understanding particles with spin 1/2. According to all the previous discussions in this blog post, the author associates these discussions with Dirac's belt or plate trick, which may have positive significance in understanding mysterious quantum phenomena. I hope that interested readers will study this possibility in depth.
It is particularly important to emphasize that the physical connotation revealed in this article is mainly manifested in the change of material properties caused by the change of the rotation number of the closed curve composed of solid materials, which involves internal energy, external force, static work and the equivalence of power and force of the material. This is exactly what the author thinks is very important and is currently studying.
References:
[1] Mei Xiangming and Huang Jingzhi, Differential Geometry, Higher Education Press, first edition, 1981 , ISBN 13010 ● 0626
[2] Wu Daren, ed., Lectures on Differential Geometry, People’s Education Press, 4th edition , October 1981 , ISBN 13012 ● 0120
[3] Su Buqing , Hu Hesheng, et al., eds., Differential Geometry, Higher Education Press (Beijing), 2nd edition (revised edition), November 2016 , JSBN : 978-04-044722-4
[4] Keying Guan, http://arxiv.org/ftp/arxiv/papers/1311/1311.6202.pdf
[5] Keying Guan, http://arxiv.org/ftp/arxiv/papers/1312/1312.2043.pdf
[6] Keying Guan, https://arxiv.org/abs/1401.3315
[7] Keying Guan, https://arxiv.org/abs/1403.1511
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