I'm going to limit my answer to biology since that's what I know about.
This is a surprisingly hard question to answer, especially in ecological (including epidemiological) and evolutionary systems, because it's very hard to be sure that you know enough about an ecological system to prove that the dynamics you're seeing are chaotic, let alone reliably inferring the existence of bifurcations. In ecology, the most reliable examples I know of are experimental, i.e. Fussman et al. 2000 (Hopf bifurcation — I do know that's not what you're looking for) and Cushing et al. 2001 (period-doubling bifurcations).
Beyond that, I looked for reasonable-seeming papers that at least referenced a particular biological system. This is not a perfectly reliable approach, as lots of modeling papers will use an empirical system as loose "inspiration" for a purely theoretical investigation of some model or set of models.
- it has long (since the 1980s) been suggested that infectious disease epidemics can undergoing period-doubling bifurcations due to seasonal forcing or many other mechanisms (Aron and Schwartz 1984) [classically measles, with its very high $R_0$ and seasonal forcing through aggregation of school children, was the poster child for this phenomenon], although strong evidence that this is the dynamical mechanism for a particular disease is hard to pin down. Aguiar et al (2009) gives one example that involves some reasonable biology for dengue.
- Guill et al. 2011 (Neimark-Sacker, Pacific salmon) and Kamimoto et al. (period-doubling, red coral) are two papers that are based at least loosely on evidence from real ecological systems, and use models to illustrate the plausibility of certain bifurcations
- chaos and bifurcations are a little easier to pin down for within-organism dynamics, because the systems are more controllable/easier to experiment on/the models may be a little more tightly linked to the actual biology (although this may be my ignorance speaking), e.g. Ryashko and Slepukhina (2017) mention Neimark-Sacker bifurcation in the context of stochastic neuronal dynamics; Quail et al 2015 discuss period-doubling in (stochastic) cardiac systems.
- two other examples of putative period-doubling bifurcations: cAMP waves/communication in slime models (Dictyostelium) (Goldbeter and Martiel 1987) and katydid chirping (stridulation) (Sismondo 1990).
- Aguiar, Maíra, Nico Stollenwerk, and Bob W. Kooi. “Torus Bifurcations, Isolas and Chaotic Attractors in a Simple Dengue Fever Model with ADE and Temporary Cross Immunity.” International Journal of Computer Mathematics 86, no. 10–11 (November 1, 2009): 1867–77. https://doi.org/10.1080/00207160902783532.
- Aron, Joan L., and Ira B. Schwartz. “Seasonality and Period-Doubling Bifurcations in an Epidemic Model.” Journal of Theoretical Biology 110, no. 4 (October 21, 1984): 665–79. https://doi.org/10.1016/S0022-5193(84)80150-2.
- Cushing, J. M., Shandelle M. Henson, Robert A. Desharnais, Brian Dennis, R. F. Costantino, and Aaron King. “A Chaotic Attractor in Ecology: Theory and Experimental Data.” Chaos, Solitons & Fractals, Chaos in Ecology, 12, no. 2 (January 2, 2001): 219–34. https://doi.org/10.1016/S0960-0779(00)00109-0.
- Fussmann, G., S. P. Ellner, K. W. Shertzer, and Jr N. G. Hairston. “Crossing the Hopf Bifurcation in a Live Predator-Prey System.” Science 290 (2000): 1358–60.
- Goldbeter, A., and J. L. Martiel. “Periodic Behaviour and Chaos in the Mechanism of Intercellular Communication Governing Aggregation of Dictyostelium Amoebae.” In Chaos in Biological Systems, edited by H. Degn, A.V. Holden, and L.F. Olsen, 79–89. Springer, 1987.
- Guill, Christian, Barbara Drossel, Wolfram Just, and Eddy Carmack. “A Three-Species Model Explaining Cyclic Dominance of Pacific Salmon.” Journal of Theoretical Biology 276, no. 1 (May 7, 2011): 16–21. https://doi.org/10.1016/j.jtbi.2011.01.036.
- Kamimoto, Sayomi, Hye Kyung Kim, Evelyn Sander, and Thomas Wanner. “A Computer-Assisted Study of Red Coral Population Dynamics.” arXiv, September 30, 2020. http://arxiv.org/abs/2008.08011.
- Quail, Thomas, Alvin Shrier, and Leon Glass. “Predicting the Onset of Period-Doubling Bifurcations in Noisy Cardiac Systems.” Proceedings of the National Academy of Sciences 112, no. 30 (July 28, 2015): 9358–63. https://doi.org/10.1073/pnas.1424320112.
- Ryashko, Lev, and Evdokia Slepukhina. “Noise-Induced Torus Bursting in the Stochastic Hindmarsh-Rose Neuron Model.” Physical Review E 96, no. 3 (September 14, 2017): 032212. https://doi.org/10.1103/PhysRevE.96.032212.
- Sismondo, Enrico. “Synchronous, Alternating, and Phase-Locked Stridulation by a Tropical Katydid.” Science 249, no. 4964 (July 6, 1990): 55–58. https://doi.org/10.1126/science.249.4964.55.