Solving bifurcation diagrams using fixed points as control parameters

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by  David Andrieux

Solving bifurcation diagrams using fixed points as control parameters
David Andrieux
Weproposetodeterminethebifurcationdiagramsoffixedpointsusingtheircoordinatesascontrol
parameters. Thismethodcanleadtoexactsolutionstootherwiseintractablebifurcationproblems.
EXECUTIVE SUMMARY
3
1 Weshowthatbifurcationsdiagramscanbeobtainedbyusingthedynamicalvariablesascontrolparameters. Instead
0 of looking at the evolution of a fixed point as a function of the system parameters, we deduce the system parameters
2
as a function of the fixed point coordinates.

n Thisapproachhasseveraladvantages. Inmanycasesanalyticalsolutionscanbeobtainedasitbypasses the need to
a solve a nonlinear system. Furthermore, by construction, it provides a global analysis. The whole bifurcation diagram
J
is obtained at one go.

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The key insight behind this result is that the system parameters are usually coupled linearly to the dynamics.
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Therefore, given a fixed point, the bifurcation problem is reduced to solving a set of linear equations for the system

parameters. The whole bifurcation diagram is then constructed by varying the fixed point coordinates.
]
h Weintroducethetechniquethroughexamples. Asimilarreasoningcanthenbeappliedtootherdynamicalsystems
c of interest.
e
m
- I. EXAMPLE 1. FITZHUGH-NAGUMO MODEL
t
a
t
s Our first example is the FitzHugh-Nagumo model
.
t
a dv
m = v−v3−w+I (1a)
dt
- dw
d τ = v−aw. (1b)
n dt
o
The FitzHugh-Nagumo describes a prototype of an excitable system (e.g., a neuron).
c
[ We use this simple example to illustrate how to simplify the bifurcation analysis by avoiding solving a cubic

nonlinearity for each values of the parameters I and a.
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The key observation is that I and a are coupled linearly to the dynamical variables v and w. Therefore, a fixed
v
point (v¯,w¯) will impose a set of linear constraints for the parameters (I,a). Solving this linear system leads to
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1
5 I(v¯,w¯) = w¯−v¯+v¯3 (2a)
1 a(v¯,w¯) = v¯/w¯. (2b)
.
1
0 These equations give the combination (I,a) for which (v¯,w¯) is a fixed point of (1). We can now obtain the whole
3 bifurcationdiagramanalyticallybyvaryingthefixedpoint(v¯,w¯)andcalculatingitsstability. Notably,thisisachieved
1 by solving a linear system rather a cubic nonlinearity as in the standard approach.
:
v
i
X II. EXAMPLE 2. BISTABILITY IN POPULATION NETWORKS
r
a
We now consider a system of two variables x ∈[0,1] interacting through a positive, increasing function F:
1,2
dx x
1 = − 1 +γ(1−x )F (W x +W x ) (3a)
dt τ 1 11 1 12 2
dx x
2 = − 2 +γ(1−x )F (W x +W x ) , (3b)
dt τ 2 21 1 22 2
where the coupling coefficients satisfy
W =W and W =W . (4)
11 22 12 21
2
Such dynamical systems appear, for example, in mean-field descriptions of neural networks (see, e.g., Ref.1).
This system admits at least one symmetric fixed point (x ,x ). Depending on the coupling strengths, the network
0 0
can also admit asymmetric states. However, none of the fixed points (symmetric or asymmetric) can be obtained
analytically.
We use our method to study the bistability properties of the network (3) in terms of the coefficients W . The
ij
interaction between the dynamical variables x and the Ws is linear. Therefore we can determine the Ws compatible
i
with a fixed point (x¯ ,x¯ ). Noting that F is invertible since it is increasing, we have that
1 2
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
x¯ x¯ W K
1 2 11 = 1 ,
x¯ x¯ W K
2 1 12 2
where
(cid:18) (cid:19)
x¯
K =F−1 i .
i γτ(1−x¯ )
i
The solution of this system reads
x¯ K −x¯ K
W (x¯ ,x¯ )=W (x¯ ,x¯ ) = 1 1 2 2 (5a)
11 1 2 22 1 2 (x¯ +x¯ )(x¯ −x¯ )
1 2 1 2
x¯ K −x¯ K
W (x¯ ,x¯ )=W (x¯ ,x¯ ) = 1 2 2 1 (5b)
12 1 2 21 1 2 (x¯ +x¯ )(x¯ −x¯ )
1 2 1 2
when x¯ (cid:54)=x¯ .
1 2
Equations (5) express the coupling parameters in terms of the fixed point coordinates (x¯ ,x¯ ). We can now obtain
1 2
the whole bistability region analytically by varying these coordinates. This bistability region is depicted in Fig 1.
FIG.1: Bistability region of the population network (3). Thecoloredregiondepictsthecouplingsforwhichsymmetric
and asymmetric stable fixed points coexist. The color code indicates the value max(x¯ ,x¯ ) of the asymmetric state. The
1 2
function F as well as the values of γ and τ used in this figure can be found in Ref.1. The coupling coefficients are determined
by Eqs. (5).
Note that, in this example, the standard problem of finding the fixed points as a function of the coupling is not
analytically tractable. Nonetheless, our approach allows us to obtain the complex bifurcation diagram seen in Fig 1
exactly. Furthermore, it gives the global solution, i.e. the whole diagram, at one go. In particular, there is no need
to follow different fixed points or branches to reconstruct the diagram.
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Acknowledgments. Part of this work was done at Yale University.
Disclaimer. This paper is not intended for journal publication.
1 K-F Wong and X-J Wang, J. Neurosci. 26, 1314 (2006).