Can Lyapunov exponent predict critical transitions in biological systems? (original) (raw)
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Chaos and Complexity Letters, 2022
The maximum Lyapunov exponent permits to know whether or not an allegedly chaotic event is in fact chaotic and, its magnitude gives an idea of its degree of chaos. The plotting of the Lyapunov exponent during a chaotic event yields information on its behavior as time evolves, however sometimes this plotting is so dense that it results in a dark spot, impossible to understand, some other times the plotting is so free of ups and downs that it seems to be nothing distinctive is happening with the flow during the event. The solution to avoid the mentioned drawbacks is making the plotting at a pre-defined time interval. The question then is how to select a convenient plotting interval so that the graph reveals more details of the behavior of the Lyapunov exponents as the system evolves. This paper shows the results of a research dealing with the above mentioned situation and, it has been encountered that a too large interval results in a neat plotting but when using a rather smaller interval, the plotting is not neat but it reveals interesting details like a pattern that is repeated along the flow. This article makes evident that selecting the adequate time interval is important to get acquainted with the Lyapunov exponent evolution. Additionally this report shows the evolution of the Lyapunov exponent for every time higher viscosity, in this case some intuitively expected comportment of the exponent has been found.
Testing Critical Slowing Down as a Bifurcation Indicator in a Low-Dissipation Dynamical System
Physical Review Letters
We study a two-dimensional low-dissipation dynamical system with a control parameter that is swept linearly in time across a transcritical bifurcation. We investigate the relaxation time of a perturbation applied to a variable of the system and we show that critical slowing down may occur at a parameter value well above the bifurcation point. We test experimentally the occurrence of critical slowing down by applying a perturbation to the accessible control parameter and we find that this perturbation leaves the system behavior unaltered, thus providing no useful information on the occurrence of critical slowing down. The theoretical analysis reveals the reasons why these tests fail in predicting an incoming bifurcation.
Using Lyapunov exponents to predict the onset of chaos in nonlinear oscillators
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
An analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed. The method is based on the Lyapunov-type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the Lyapunov characteristic exponents. The concept of temporally local Lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable, and, therefore, the existence of chaotic attractors is impossible. The procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point. By considering the inverse conditions of asymptotic stability, this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems, including many cases of practical interest. The examples of externally excite...
Critical transitions and perturbation growth directions
Physical Review E, 2017
Critical transitions occur in a variety of dynamical systems. Here, we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for critical transitions, we consider changes in growth rates and directions of covariant Lyapunov vectors. Studying critical transitions in several models of fast-slow systems, i.e., a network of coupled FitzHugh-Nagumo oscillators, models for Josephson junctions and the Hindmarsh-Rose model, we find that tangencies between covariant Lyapunov vectors are a common and maybe generic feature during critical transitions. We further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operating characteristic curves. In the presence of noise, we find the alignment of covariant Lyapunov vectors and changes in finite-time Lyapunov exponents to be more successful in announcing critical transitions than common indicator variables as, e.g., finite-time estimates of the variance. Additionally, we propose a new method for estimating approximations of covariant Lyapunov vectors without knowledge of the future trajectory of the system. We find that these approximated covariant Lyapunov vectors can also be applied to predict critical transitions.
Universal Behavior of Lyapunov Exponents in Unstable Systems
Physical Review Letters, 1995
We calculate the Lyapunov exponents in a classical molecular dynamics framework. Yukawa and Slater-Kirkwood forces are considered in order to give an equation of state that resembles the nuclear and the atomic 4 He equation of state, respectively, near the critical point for liquid-gas phase transition. The largest Lyapunov exponents l are always positive and can be very well fitted near the "critical temperature" with a functional form l~jT 2 T c j 2v , where the exponent v 0.15 is independent of the system and mass number. At smaller temperatures we find that l~T 0.4498 , a universal behavior characteristic of an order to chaos transition.
Lyapunov Exponents, Singularities, and a Riddling Bifurcation
Physical Review Letters, 1997
There are few examples in dynamical systems theory which lend themselves to exact computations of macroscopic variables of interest. One such variable is the Lyapunov exponent which measures the average attraction of an invariant set. This article presents .a family of noninvertible transformations of the plane for which such computations are possible. This model sheds additional insight into the notion of what it can mean for an attracting invariant set to have a riddled basin of attraction.
Determining Lyapunov exponents from a time series
Physica D-nonlinear Phenomena, 1985
We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor.