It is known that stability properties of delay-differential equations are not preserved by Taylor series expansion of the delayed term. Still, this technique is often used to approximate delayed systems by ordinary differential equations in different engineering and biological applications. In this brief, it is demonstrated through some simple second-order scalar systems that low-order Taylor series expansion of the delayed term approximates the asymptotic behavior of the original delayed system only for certain parameter regions, while for high-order expansions, the approximate system is unstable independently of the system parameters.

References

References
1.
Orosz
,
G.
,
Wilson
,
R. E.
, and
Stepan
,
G.
,
2010
, “
Traffic Jams: Dynamics and Control
,”
Philos. Trans. R. Soc., A
,
368
(
1928
), pp.
4455
4479
.10.1098/rsta.2010.0205
2.
Mao
,
X.
, and
Hu
,
H.
,
2010
, “
Stability and Bifurcation Analysis of a Network of Four Neurons With Time Delays
,”
ASME J. Comput. Nonlinear Dynam.
,
5
(
4
), p.
041001
.10.1115/1.4000317
3.
Szalai
,
R.
, and
Orosz
,
G.
,
2013
, “
Decomposing the Dynamics of Heterogeneous Delayed Networks With Applications to Connected Vehicle Systems
,”
Phys. Rev. E
,
88
(
4
), p.
040902
.10.1103/PhysRevE.88.040902
4.
Erneux
,
T.
, and
Kalmar-Nagy
,
T.
,
2007
, “
Nonlinear Stability of a Delayed Feedback Controlled Container Crane
,”
J. Vib. Control
,
13
(
5
), pp.
603
616
.10.1177/1077546307074245
5.
Ma
,
H.
, and
Butcher
,
E. A.
,
2005
, “
Stability of Elastic Columns With Periodic Retarded Follower Forces
,”
J. Sound Vib.
,
286
, pp.
849
867
.10.1016/j.jsv.2004.10.052
6.
Takacs
,
D.
, and
Stepan
,
G.
,
2009
, “
Experiments on Quasiperiodic Wheel Shimmy
,”
ASME J. Comput. Nonlinear Dynam.
,
4
(
3
), p.
031007
.10.1115/1.3124786
7.
Bachrathy
,
D.
,
Turi
,
J.
, and
Stepan
,
G.
,
2011
, “
State Dependent Regenerative Effect in Milling Processes
,”
ASME J. Comput. Nonlinear Dynam.
,
6
, p.
041002
.10.1115/1.4003624
8.
Insperger
,
T.
, and
Stepan
,
G.
,
2011
,
Semi-Discretization for Time-Delay Systems
,
Springer
,
New York
.
9.
Bobrenkov
,
O. A.
,
Nazari
,
M.
, and
Butcher
,
E. A.
,
2012
, “
Response and Stability Analysis of Periodic Delayed Systems With Discontinuous Distributed Delay
,”
ASME J. Comput. Nonlinear Dynam.
,
7
(
3
), p.
031010
.10.1115/1.4005925
10.
Èl'sgol'c
,
L.
,
1964
,
Qualitative Methods in Mathematical Analysis
,
American Mathematical Society
,
Providence, RI
.
11.
Mazanov
,
A.
, and
Tognetti
,
K. P.
,
1974
, “
Taylor Series Expansion of Delay Differential Equations—A Warning
,”
J. Theor. Biol.
,
46
(
1
), pp.
271
282
.10.1016/0022-5193(74)90152-0
12.
Driver
,
R.
,
1977
,
Ordinary and Delay Differential Equations
,
Springer-Verlag
,
New York
.
13.
Kurzweil
,
J.
,
1971
, “
Small Delays Don't Matter
,”
Proceedings of the Symposium on Differential Equations and Dynamical Systems, Lecture Notes in Mathematics
, D. Chillingworth, ed., Springer, pp.
47
49
.
14.
Driver
,
R. D.
,
Sasser
,
D. W.
, and
Slater
,
M. L.
,
1973
, “
The Equation x'(t)=ax(t)+bx(t-τ) With “Small” Delay
,”
Am. Math. Mon.
,
80
(
9
), pp.
990
995
.10.2307/2318773
15.
Guillouzic
,
S.
,
L'Heureux
,
I.
, and
Longtin
,
A.
,
1999
, “
Small Delay Approximation of Stochastic Delay Differential Equations
,”
Phys. Rev. E
,
59
(
4
), pp.
3970
3982
.10.1103/PhysRevE.59.3970
16.
Morrison
,
T. M.
, and
Rand
,
R. H.
,
2007
, “
2:1 Resonance in the Delayed Nonlinear Mathieu Equation
,”
Nonlinear Dyn.
,
50
(
1–2
), pp.
342
352
.10.1007/s11071-006-9162-5
17.
Stepan
,
G.
, and
Kollar
,
L.
,
2000
, “
Balancing With Reflex Delay
,”
Math. Comput. Model.
,
31
(
4–5
), pp.
199
205
.10.1016/S0895-7177(00)00039-X
18.
Asai
,
Y.
,
Tasaka
,
Y.
,
Nomura
,
K.
,
Nomura
,
T.
,
Casidio
,
M.
, and
Morasso
,
P.
,
2009
, “
A Model of Postural Control in Quiet Standing: Robust Compensation of Delay–Induced Instability Using Intermittent Activation of Feedback Control
,”
PLoS ONE
,
4
,
e6169
.10.1371/journal.pone.0006169
19.
Qu
,
X.
,
Nussbaum
,
M. A.
, and
Madigan
,
M. L.
,
2007
, “
A Balance Control Model of Quiet Upright Stance Based on an Optimal Control Strategy
,”
J. Biomech.
,
40
(
16
), pp.
3590
3597
.10.1016/j.jbiomech.2007.06.003
20.
Paoletti
,
P.
, and
Mahadevan
,
L.
,
2012
, “
Balancing on Tightropes and Slacklines
,”
J. R. Soc. Interface
,
9
(
74
), pp.
2097
2108
.10.1098/rsif.2012.0077
21.
Stepan
,
G.
,
1989
,
Retarded Dynamical Systems
,
Longman Scientific & Technical and John Wiley & Sons
,
NY
.
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