Abstract
Basin stability (BS) is a universal concept for complex systems studies, which focuses on the volume of the basin of attraction instead of the traditional linearizationbased approach. It has a lot of applications in realworld systems especially in dynamical systems with a phenomenon of multistability, which is even more ubiquitous in delayed dynamics such as the firing neurons, the climatological processes and the power grids. Due to the infinite dimensional property of the space for the initial values, how to properly define the basin’s volume for delayed dynamics remains a fundamental problem. We propose here a technique which projects the infinite dimensional initial state space to a finitedimensional Euclidean space by expanding the initial function along with different orthogonal or nonorthogonal basis. A generalized concept of basin’s volume in delayed dynamics and a highly practicable calculating algorithm with a crossvalidation procedure are provided to numerically estimate the basin of attraction in delayed dynamics. We show potential applicabilities of this approach by applying it to study several representative systems of biological or/and physical significance, including the delayed Hopfield neuronal model with multistability and delayed complex networks with synchronization dynamics.
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Introduction
The linear stability theory plays a central role in complex systems science, which describes the longterm behavior of system’s steady states, such as equilibriums and periodic orbits, after a small perturbation. Such a linearizationbased approach has many applications both theoretically and practically by determining whether the steady state is stable or not (in terms of the sign of the Lyapunov exponents)^{1}. However, due to the localness of this approach, it is only valid when the perturbation is limited to a small range. Since nonsmall perturbations are omnipresent in realworld systems, a global method for complementing the linearstability paradigm needs to be developed. Although the classical Lyapunov function method with the Invariance Principle could be a candidate^{2,3,4,5,6}, this method, which usually requires subtle construction techniques and highly depends on the accurate form of the systems investigated, has some theoretical significance but lacks of extensively practical usefulness. Fortunately, the basin stability (BS), proposed recently by Menck et al., becomes such a global method, which uses the basin of attraction instead to fully assess the steady state’s stability^{7}. This method focuses on the basin’s volume which serves as a measure of the likelihood of return to a steady state after being subjected to any random perturbation^{8} and thus explains to what extent a steady state is stable in a probability sense.
The BS approach has become a powerful tool for complex systems studies and has already been successfully applied to many fields of science^{7,9,10,11}, especially in those systems with multistability. As a matter of fact, the phenomenon of multistability is even more ubiquitous in delayed dynamical systems^{9,12,13,14}. Moreover, due to a certain amount of time duration for the signal transmission, such as the finite switching speed and the memory effect, time delay exists in almost every real system^{15,16,17,18}. Therefore, it is of great significance to consider the BS in delayed dynamics. To this end, we need to define the basin’s volume in a timedelayed sense, while the space for the initial values of a timedelayed differential equation (DDE) is a function space which has dimension of infinity^{19}. Although by measure theory, it is possible to construct analytical measures, such as the Hausdorff measure, to assess the basin’s ‘volume’^{20}, such measures are less geometric intuitive or numerical computable as basin’s area or volume in the traditional BS theory of ordinary differential equations.
To overcome these difficulties, we articulate in the following the concept of the BS by a technique which projects the infinite dimensional initial state space of the DDE to a finite dimensional Euclidean space. A function of the initial value is related to its coefficients under the expanding along with an orthogonal or a nonorthogonal basis. Through this, a generalized notion of basin’s volume is theoretically described and a practical calculating algorithm with a procedure of crossvalidation is introduced. We then apply the developed approach to study a multistable Hopfield neuronal model with time delays. Some further applications, including the investigations of the synchronization of delayed complex networks and multistability of equilibriums and periodic orbits in timedelayed oscillators, show the broad potential of applications of this developed approach.
Results
Consider a system described by the delayed differential equation
where x is an sdimensional state variable, for and τ is a time delay. The set, denoted by , for the initial values of system (1) is a space consisting of continuous functions. We use to represent the Euclidean norm in an ndimensional. Now, we state the definition of basin of attraction for a stable steady state, which could be either an equilibrium or a periodic orbit.
Basin of attraction for delayed dynamics
Suppose is an isolated asymptotically stable equilibrium or periodic orbit of system (1). If there exists a region for initial values such that for any , the solution of the system starting from z, denoted by , uniformly converges to ϕ^{*} as t → +∞, i.e. , then is called the basin of attraction (or basin) with respect to the stable steady state ϕ^{*}.
Because the function space for the initial values is taken into account, the basin of attraction of system (1) is of infinite dimension. A direct computation for measuring this kind of BS is impossible, so that some approximate algorithm should be adopted. Notice that, for any and a given standard orthogonal basis , there exists a unique sequence of numbers such that
where the limit is interpreted in a sense of the square integral on and are the Fourier coefficients with ^{21}. Refer to the section of Method for specific bases that we use in the paper. We thus construct a set through , where α is a sufficiently large positive constant and . Due to the onetoone property between C_{(n,α)} and the bounded set in , we use the conventional Lebesgue measure m to count the volume of C_{(n,α)} as . We call the volume of the geometry the nthorder generalized basin’s volume with respect to the given standard orthogonal basis (also called basin’s volume if not confused). We further normalize this volume through
and denote this normalized value by . is then used to represent the nthorder approximate BS of the basin for the delayed dynamics (1).
For the sake of practical use, we introduce how to numerically compute the generalized BS defined above. We integrate system (1) for T points drawn uniformly at random from , which correspond to T initial functions [a point is mapped to the function g_{n} defined in (2)] and then count the number M of the initial functions that finally arrive at the stable steady state ϕ^{*}. can then be estimated as M/T when T is sufficiently large. In Supplementary Materials, we analytically show how to choose a proper value of T for getting a credible estimation on the nthorder approximate BS.
Moreover, the generalized BS, as defined in (3), depends on both n and the ranges of the coefficients a_{i}. Analytical proof of the BS convergence with increasing either n or the range is hard at present. However, numerical validation of this convergence could be adopted in applications, which is shown in the following examples as well as the examples in Supplementary Materials. In fact, when the fluctuations of , persistently for larger n’s or wider ranges, are below an allowable bias specified a priori, we take the smallest value of such n’s as the number of the basis functions that are successively taken from the standard orthogonal basis and take the smallest value of such boundaries for the coefficient a_{i} used in C_{(n,α)}.
Clearly, the above procedure for defining and computing the generalized BS for delayed dynamics could be performed not only with the orthogonal bases, such as the trigonometric and the Legendre bases (refer to the section of Method), but also with the nonorthogonal bases, including the Bernstein basis^{22} (refer to Supplementary Materials). Then, in order to avoid losing the essential functions for the initial values in computation, we determine n and the ranges for the coefficients through a crossvalidation procedure in applications, that is, they take values such that the aforementioned convergence requirements on the approximate BS are satisfied simultaneously for three different bases. Interestingly, as shown in the following examples, although the exact values of the BS for different bases are not uniform, they display almost the same changing tendency with the variation of some parameters such as the time delay.
Delayed Hopfield neuronal model
We first apply the generalized BS approach to the delayed neuronal model. Multistability exists widely in neural networks especially when applied to associative memory^{23,24}. Several sample patterns (represented as equilibriums) are stored in the weights of networks and associative memory is used to recover from an incomplete or damaged version to its complete pattern with a help of network dynamics. The number of equilibriums represents the capacity of memorial storage and the basin’s volume is the error correcting capability that determines which memory is of the highest possibility to be remembered. By our approach, we can estimate the volume of several patterns relatively and discover its relationship with the neuron’s parameters. We demonstrate it by considering a simple twodimensional delayed Hopfield neuronal model^{25}, described by
where u_{i}(t) is the state of neuron i at time t. Parameters a, b, a_{12}, a_{21}, I_{1} and I_{2} are constants with a > 0, b > 0 and a_{ij} ≥ 0 for i, j = 1, 2 and i ≠ j. is a nonlinear neuronal activation function. Here, the parameters are set as a = 2, a_{12} = a_{21} = 0.55, b = 1, I_{1} = I_{2} = 0 and f(x) = tanh(x). Time delays in the equations represent the transmission times between different neurons and here we let τ_{s} = τ_{12} = τ_{21} = τ. By linear stability theory, the model (4) with has four asymptotically stable equilibrium points, i.e., , , and , as shown in Fig. 1(a,d,g). For τ > 0, these equilibriums correspond to four constant function vectors on the interval [−τ, 0], respectively. In the meanwhile, looking into the BS reveals us more interesting detail information about these equilibriums besides whether they are stable or not.
First, in order to find the appropriate number of basis functions in calculating the nthorder approximate BS for the four equilibrium points, we use the algorithms with a crossvalidation procedure introduced above. In particular, take in account the Bernstein, the trigonometric and the Legendre bases simultaneously, set α = 6 and denote by for simplicity. As shown in Fig. 1(b,e,h), the fluctuations of for every basis tend to be stable when n is sufficiently large and thus for the model (4), we take n = 20 as the number of the basis functions that are successively selected from the given bases. Since represents the possibility of a pattern that might be remembered, investigating the variation of its value with some parameters becomes an important issue. As shown in Fig. 1(c,f,i), values of for the two equilibriums E_{1,2} are getting smaller and smaller when the time delay τ increases (showing a less probability of being remembered) and finally the two other patterns, E_{3,4}, become dominant with an almost equal proportion. For all the three bases, Fig. 1(j) depicts the normalized curves of the variation of with τ for the attractor E_{1}. As mentioned above, despite the fact that values of for different bases are not exactly uniform, the variations of these values with the time delay approach a high consensus.
While human brains have a large amount of neurons, the capacity of memorial storage is even enormous. As shown by the example, the BS provides us a potential method to fully discover the multiattractors and their basins. And comparing to the linearization approach, the BS proves itself a practical tool which provides a global view of the system in a probability sense.
Application to complex networks with timedelayed couplings
To show the applicability of this approach even to delayed complex networks, we apply it to study the synchronization phenomenon here. Synchronization is a fundamental nonlinear behavior^{26}, which can be observed in many systems such as the firing neurons^{13}, the human cardiorespiratory system^{27}, the singing crickets and the power grids^{9}. The original BS theory shows us to what extent a system will maintain synchronous behavior when perturbations occur and the relationship between networks’ topology and the basin’s volume. An amazing discovery is that the BS of a network shows a quite different trend versus the WattsStrogatz network’s parameter p, compared to the traditional indicator, synchronizability R, which is always used to measure the network’s capability of reaching synchronization^{7}.
We thus investigate the generalized BS of the complex networks where we take into account the time delay, which may affect the system’s stability of synchronization or even change the synchronization manifold^{28,29}. We consider an undirected complex network with N delaycoupled identical nodes, which is represented as follows^{30}:
where and . The dynamics of the individual units is governed by the function F, σ denotes the overall coupling strength constant and the network’s Laplacian G = g_{ij} is an N × N matrix which determines the topology of the network. h is a coupling function (we choose it as an identity map here) and τ is the coupling delay. Under the requirement of , system (5)’s synchronization manifold is . By considering the largest transverse Lyapunov exponent of the master stability function (MSF)^{31}, for a large amount of networks we can obtain a range of the normalized coupling parameter ε which depends on coupling strength σ and coupling matrix G (see Methods). In this range, system (5) can be stably synchronized. In the following discussion, the WattsStrogatz networks consisting of the paradigmatic Rössler oscillators are taken into account^{32}. A single Rössler oscillator with given parameters is governed by
Here, the x → x component coupling is used to build the network^{30}. According to the MSF method, the stable synchronous state exists only in the range of the normalized coupling parameter in the absence of the coupling delay. As τ increases to 0.24, a second stable regime ε > 7.38 emerges and when τ reaches 0.4, again only one stable regime exists with ^{33}. It is easy to calculate the range of σ according to ε and different network topology G (see Methods). In the following we concentrate on the regime τ = 0.4. The rewiring probability p in the WattsStrogatz network (p = 0 for regular lattices and p = 1 for random graphs) is varied to see the relationship between network topology and its synchronization.
From the traditional viewpoint, network’s synchronizability R, defined as γ_{max}/γ_{min}, the ratio of matrix G’s maximum and minimum nonzero eigenvalues (the smaller R is, the more synchronizable the network is), always indicates an improvement with the increasing of p (from unstable regime to stable regime), see the blue dashed curve plotted in Fig. 2. However, akin to the results for the nondelayed networks^{7}, the mean basin stability (see Methods for details) shoots up when p is small and reaches a peak. After that, just opposite to the variation of R with p, an exponential decline of the mean basin stability can be observed, which means the network becomes more synchronizable but the basin of the synchronization manifold is otherwise decreasing. Clearly, this application of the generalized BS approach on delayed networks also complements the linear synchronization studies.
Moreover, this generalized BS approach could also be applied to systems with periodic orbits or even chaotic attractors, which would reveal more subtle nonlinear properties (refer to Supplementary Materials for examples including the delayed Van der PolDuffing oscillator model and the delay coupled StuartLandau system). All these examples show that the generalized BS approach for delayed dynamics could provide a relatively comprehensive knowledge of the basin of the steady state and shows whether a perturbation is permissible, which is quite essential in engineering.
Conclusions
The linear stability theory provides a local method to determine whether a state is stable or not. However, limited by the infinite dimension of the initial state space, it is still difficult to have a comprehensive understanding of attractors in delayed dynamics. Although the method of LyapunovKrasovskii functional could be used to analyze the multistability and basins of multiattractors, this method still produces a sufficient condition, lacking a capacity of describing the exact shape of an attractor’s basin^{4,19} and also this method in essence still restricts the discussion in the finitedimensional space, describing the condition on initial value only at a time of zero^{34,35}. To overcome these difficulties, we propose here an approximate approach of applying basin stability to delayed dynamics. The BS approach turns our eyes from a traditional local linearized view of stability to a global inspection. And our method extends its application and converts an infinite dimensional problem into a highly practicable algorithm especially when time delay plays an important role in realworld systems. The generalized BS approach even provides an opportunity to fully investigate more nonlinear properties of attractors in delayed dynamics such as the bifurcation phenomena. Furthermore, an analytical proof on the convergence of our proposed approach for computing the generalized BS is also of great significance, which becomes a topic of our present and future works.
Methods
Orthogonal bases
We take, respectively, the orthogonal basis consisting of trigonometric functions on [−π, π] as follows:
and the Legendre basis on [−1, 1], which is also an orthogonal basis, as follows:
In the framework of functional analysis, a series, which is generated by expanding a function along with a given standard orthogonal basis, could be regarded as a Fourier series^{21}. Moreover, in order to realize the crossvalidation procedure formalized in the section of Result, we also use the nonorthogonal Bernstein basis, which is introduced in Supplementary Materials.
Remember that different basis leads to different . So we should make clear which basis is used if we need to focus on the exact value of . As shown in the above examples and examples in Supplementary Materials, almost the same results on the changing tendency can be obtained while considering the variation of the generalized basin’s volume with some parameters.
Master stability function
The stability of the synchronization manifold is determined by the eigenvalues of the system’s Laplacian G and the following variational equation (i.e., MSF)^{31}
Here, ξ is an mdimensional variation transverse to the synchronization manifold. The normalized coupling parameter ε = σγ_{k}, in which γ_{k} denotes the eigenvalues of G for . Thus, the stability of the synchronized state of a given network topology G is guaranteed by the negative sign of the largest transverse Lyapunov exponent of Eq. (6) ^{30}.
Estimation of mean basin stability
Notice that the stable synchronous state requires σ to take its value in a specific range. Here, we estimate the basin stability for several different equally spaced values in this range and average to obtain a mean . Additionally, several networks are produced for one probability p to acquire the average value of .
Additional Information
How to cite this article: Leng, S. et al. Basin stability in delayed dynamics. Sci. Rep. 6, 21449; doi: 10.1038/srep21449 (2016).
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Acknowledgements
S.L. was supported by the China Scholarship Council (CSC) scholarship (Grant No. 501100004543). W.L. was supported by the National Natural Science Foundation (NNSF) of China (Grants No. 61273014 and No. 11322111).
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W.L. conceived the idea and S.L. performed the research and analyzed the data. W.L., S.L. and J.K. wrote the paper.
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Leng, S., Lin, W. & Kurths, J. Basin stability in delayed dynamics. Sci Rep 6, 21449 (2016). https://doi.org/10.1038/srep21449
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DOI: https://doi.org/10.1038/srep21449
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