37 research outputs found
Bogdanov–Takens and triple zero bifurcations in general differential systems with m delays
This paper mainly concerns the derivation of the normal forms of the Bogdanov–Takens (BT) and triple zero bifurcations for differential systems with m discrete delays. The feasible algorithms to determine the existence of the corresponding bifurcations of the system at the origin are given. By using center manifold reduction and normal form theory, the coefficient formulas of normal forms are derived and some examples are presented to illustrate our main results
A mathematical framework for critical transitions: normal forms, variance and applications
Critical transitions occur in a wide variety of applications including
mathematical biology, climate change, human physiology and economics. Therefore
it is highly desirable to find early-warning signs. We show that it is possible
to classify critical transitions by using bifurcation theory and normal forms
in the singular limit. Based on this elementary classification, we analyze
stochastic fluctuations and calculate scaling laws of the variance of
stochastic sample paths near critical transitions for fast subsystem
bifurcations up to codimension two. The theory is applied to several models:
the Stommel-Cessi box model for the thermohaline circulation from geoscience,
an epidemic-spreading model on an adaptive network, an activator-inhibitor
switch from systems biology, a predator-prey system from ecology and to the
Euler buckling problem from classical mechanics. For the Stommel-Cessi model we
compare different detrending techniques to calculate early-warning signs. In
the epidemics model we show that link densities could be better variables for
prediction than population densities. The activator-inhibitor switch
demonstrates effects in three time-scale systems and points out that excitable
cells and molecular units have information for subthreshold prediction. In the
predator-prey model explosive population growth near a codimension two
bifurcation is investigated and we show that early-warnings from normal forms
can be misleading in this context. In the biomechanical model we demonstrate
that early-warning signs for buckling depend crucially on the control strategy
near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio
Non-reciprocal phase transitions
Out of equilibrium, the lack of reciprocity is the rule rather than the
exception. Non-reciprocal interactions occur, for instance, in networks of
neurons, directional growth of interfaces, and synthetic active materials.
While wave propagation in non-reciprocal media has recently been under intense
study, less is known about the consequences of non-reciprocity on the
collective behavior of many-body systems. Here, we show that non-reciprocity
leads to time-dependent phases where spontaneously broken symmetries are
dynamically restored. The resulting phase transitions are controlled by
spectral singularities called exceptional points. We describe the emergence of
these phases using insights from bifurcation theory and non-Hermitian quantum
mechanics. Our approach captures non-reciprocal generalizations of three
archetypal classes of self-organization out of equilibrium: synchronization,
flocking and pattern formation. Collective phenomena in these non-reciprocal
systems range from active time-(quasi)crystals to exceptional-point enforced
pattern-formation and hysteresis. Our work paves the way towards a general
theory of critical phenomena in non-reciprocal matter.Comment: Supplementary movies at
https://home.uchicago.edu/~vitelli/videos.htm