INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN SOCIAL SCIENCE (IJRISS)  
ISSN No. 2454-6186 | DOI: 10.47772/IJRISS | Volume IX Issue XI November 2025  
Applying Singular Perturbation Theory for Reducing the Size of  
Dynamic Model of the Electronic Circuits  
C. F. Khoo  
Centre for Telecommunication Research and Innovation (CeTRI), Fakulti Teknologi dan Kejuruteraan  
Elektronik dan Komputer, Universiti Teknikal Malaysia Melaka  
DOI: https://dx.doi.org/10.47772/IJRISS.2025.91100506  
Received: 06 December 2025; Accepted: 12 December 2025; Published: 20 December 2025  
ABSTRACT  
The dynamic modelling of nonlinear electronic circuits often results in high-dimensional systems of differential  
equations that are computationally expensive to solve, particularly when incorporating parasitic elements with  
widely varying time scales. This research proposes a dimensionality reduction framework utilizing singular  
perturbation theory applied to the chaotic Chua circuit. By decomposing the system dynamics into "slow" (outer)  
and "fast" (inner) time scales, and invoking Tikhonov’s theorem to validate the asymptotic correctness, a  
reduced-order model is derived. A uniform approximation is subsequently constructed by mathematically  
matching the boundary layer transients with the steady-state behaviour. Numerical simulations compare this  
approximation against the full system solved via standard ODE solvers, revealing that the uniform approximation  
achieves high fidelity with negligible absolute errors. The results confirm that singular perturbation is an  
effective technique for minimizing computational cost without compromising dynamical accuracy, presenting  
significant potential for scaling to higher-dimensional problems.  
Keywords: Perturbation theory, Tikhonov's theorem, Chua circuit  
INTRODUCTION  
The dynamic modelling of electronic circuits is fundamental to understanding system behaviours, particularly  
when characterizing complex relationships between voltage, current, and nonlinear resistive elements. However,  
accurate dynamic modelling often results in high-dimensional systems of ordinary differential equations (ODEs)  
that are computationally expensive to solve, especially when integrating stiff equations with widely varying time  
scales. As noted in recent studies on spacecraft electrical systems and microgrid clusters, the presence of parasitic  
parameters—such as small inductances and capacitances—often creates multiple time scales that complicate  
numerical stability [1], [2]. To address this, singular perturbation theory (SPT) offers a robust mathematical  
framework, allowing researchers to decompose a high-dimensional system into reduced-order "slow" and "fast"  
subsystems. By mathematically isolating the boundary layer phenomena where fast transients occur, SPT  
provides a systematic method for model reduction that retains the essential dynamics of the system while  
significantly lowering computational overhead [3], [4].  
METHODOLOGY  
Mathematical Modelling  
A1.  
Chua Circuit  
The Chua circuit was first invented in 1983 by Leon O. Chua. It is a simple electronic circuit that exhibits chaos  
and many bifurcation phenomena. The existence of chaotic attractors from the Chua circuit had been confirmed  
numerically by Matsumoto via computer simulations, observed experimentally by Zhong and Ayrom in  
laboratory, and mathematically proved by Chua et al. in [5].  
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INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN SOCIAL SCIENCE (IJRISS)  
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The circuit diagram of the Chua Circuit is shown in Figure 1. The circuit consists of five circuit elements: two  
capacitors C1 and C2, one inductor L1, one linear resistor R and one nonlinear resistor NR. The component NR is  
a nonlinear negative resistance called a Chua's diode. It is usually made of a circuit containing an amplifier with  
positive feedback.  
Figure 5.1: The Chua circuit  
Recent literature highlights the evolving complexity of Chua’s circuit, extending beyond simple chaotic  
demonstration to sophisticated control and synchronization applications. Xu et al. demonstrated the utility of  
Field-Programmable Gate Arrays (FPGAs) in numerically simulating synchronized Chua circuits, emphasizing  
the need for efficient model representations to match hardware constraints [6]. Similarly, Sun et al. and  
Chaudhury et al. have explored synchronous dynamics in robotic arms and heterogeneous oscillators driven by  
Chua circuits, respectively, reinforcing the necessity of precise yet computationally manageable mathematical  
models [7], [8]. Furthermore, the investigation of Jacobi stability in Muthuswamy–Chua–Ginoux systems by  
Wang et al. illustrates that even in modified circuit topologies, the core challenges of nonlinear stability and  
dimensionality persist [9]. These studies collectively suggest that while the physical implementations of chaos  
are advancing, the demand for analytical methods that can simplify these high-dimensional interactions without  
losing topological accuracy is higher than ever.  
A2.  
Mathematical Model of Chua Circuit  
The Chua circuit can be analysed by using Kirchhoff's circuit laws, the dynamics of this circuit can be modelled  
by a system of three nonlinear ordinary differential equations (ODEs) in the variables x(t), y(t) and z(t), which  
represent the voltages across the capacitors C1 and C2, and the intensity of the electrical current in the inductor  
L1, respectively. The system of ODEs has the form  
푑푥  
푑푡  
1
휖
3
2
(
)
푧 − 푐1ꢀ − 푐2ꢀ − 휇 ꢀ  
=
푑푦  
푑푡  
= −훽ꢀ  
(1)  
푑ꢁ  
푑푡  
= −훼ꢀ + ꢂ + 푏푧  
Here the ꢃ holds a small value while the parameters c1, c2, 휇, 훽, a and b are determined by the particular values  
of the circuit components.  
A3.  
Singular Perturbation Theory  
Let consider a multiple time scales system of the form  
푑푥  
(
)
(
)
ꢃ
= 푓 ꢀ, ꢂ ,  
ꢀ 0, ꢃ = ꢀ0  
(2)  
푑푡  
푑푦  
푑푡  
(
)
(
)
= 푔 ꢀ, ꢂ ,  
ꢂ 0, ꢃ = ꢂ0  
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푛
푚
(
]
[
]
where ꢀ ∈ 푅 , ꢂ ∈ 푅 , ꢃ ∈ 0, 1 , ꢄ ∈ 0, 푇 , 푓 and 푔 are sufficiently smooth function in the domain  
(
)
퐷 = { ꢀ, ꢂ : |ꢀ| ≤ ꢅ1, |ꢂ| ≤ ꢅ2},  
ꢅ1, ꢅ2 ∈ 푅+.  
That is, 푓 ∈ 퐶푟(퐷, 푅푛) and 푔 ∈ 퐶푟(퐷, 푅푚) for ꢆ ≥ 2. The variable x is called the fast variable while variable y  
is the slow variable in the system.  
Formally, by setting ꢃ = 0, we obtain a differential algebraic system:  
0 = 푓(ꢀ  
̅
, ꢂ̅),  
(3)  
ꢅꢂ̅  
(
)
, ꢂ̅ ,  
(
)
= 푔 ꢀ  
̅
ꢂ̅ 0, 0 = ꢂ0  
ꢅꢄ  
This system is called the “degenerate system” in the singular perturbation theory as its order is less than the order  
of system (2). Since system (3) has reduced to the differential algebraic equation (DAE) form, system (2) is an  
example of a singularly perturbed system. Then Tikhonov theorem [10] is referred as it shows how well the  
system (2) is approximated by the unperturbed system (3).  
On the other hand, by using the scaled time variable 휏 = ꢄ/ꢃ, system (2) can be reformulated in an equivalent  
form:  
푑푥̃  
̃
(
)
(
)
= 푓 ꢀ̃ , ꢂ̃ ,  
ꢀ̃ 0, ꢃ = ꢀ0  
(4)  
푑푡  
푑푦̃  
푑푡  
(
)
(
)
= ꢃ푔̃ ꢀ̃, ꢂ̃ ,  
ꢂ̃ 0, ꢃ = ꢂ0  
ꢇ
~
(
]
[
]
where ꢃ ∈ 0,1 , 휏 ∈ 0, 휖  
,
and  
g
are sufficiently smooth functions in the domain  
f
̃
(
)
퐷 = { ꢀ̃, ꢂ̃ : |ꢀ̃| ≤ 푐1, |ꢂ̃| ≤ 푐2},  
푐1, 푐2 ∈ 푅+.  
This system is a regularly perturbed ODE. By referring to the regular perturbation theory, we know that the  
system (4) is well-approximated by the adjoined system [11].  
In general, two solutions (approximations) can be derived for a singularly perturbed problem when setting ꢃ =  
0. The “outer solution”, the solution for the degenerate system (3), provides a good approximation outside the  
boundary layer. On the other hand, the “inner solution”, the solution for the adjoined system (4), provides a good  
approximation in the boundary layer. A process of matching of these solutions is essential to relate the two  
solutions and obtain a complete solution that approximates the system dynamics throughout the whole problem  
domain. Kaplun and Langerstorm [12] employed “intermediate matching” which is based on the so-called  
“overlapped hypothesis”, that is, an assumption that there exist extended domains of validity for the outer and  
inner expansions with a non-empty intersection, which is where both the outer and inner expansions are valid  
and can be matched. More precisely, the inner 휑0(휏) and outer approximation 휙0(ꢄ) are matched if they have a  
common limit as ꢃ tends to zero, hence the requirement for matching can be represented as  
lim 휙0(ꢄ) = lim 휑0(휏)  
휖→0  
휖→0  
Subsequently, the final, matched solution that well-approximates the actual solution on the whole problem  
domain is called the “uniform approximation”. This approximation can be obtained by adding the inner and outer  
solutions and subtracting their common limit as follows:  
( )  
( )  
( )  
Φ푢 ꢄ = 휙0 ꢄ + 휑 휏 − ηc  
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where u (t) is the uniform approximation and  
ηc = lim 휙0(ꢄ) = lim 휑0(휏)  
ꢈ→∞  
푡→0  
is the common limit.  
Simulation  
B1.  
Outer solution  
Within the slow time scale, consider the system of ODEs (1). By letting ꢃ = 0, the first equation of system (1)  
reduces to an algebraic equation  
푧 − 푐1ꢀ표3 − 푐2ꢀ표2 − 휇 ꢀ표 = 0  
(5)  
Therefore, the voltages across the capacitors C1, xo, in the outer region, appears to be constant in the post-  
transient time course. Here, xo , yo and zo are used to replace the x, y and z respectively to denote the x, y and z in  
outer region.  
Meanwhile, rewrite the second and the third equations of the ODEs system (1) with the substitution of xo gives  
푑 푦  
ꢉ
= −훽 ꢀ표  
(6)  
푑푡  
ꢅ 푧표  
= −훼ꢀ표 + ꢂ표 + 푏푧표  
ꢅꢄ  
where xo can be found numerically from the equation (5).  
B2. Inner Solution  
푡
On the other hand, upon replacement of the scaling dimensionless variables, 휏 = into the system of equations  
휖
(1), the new governing equations can be written as  
푑푥  
퐼
= 푧 − 푐1ꢀ3 − 푐2ꢀ2 − 휇ꢀꢊ  
ꢊ
ꢊ
푑푡  
푑푦  
퐼
= −ꢃ훽ꢀꢊ  
(7)  
푑푡  
푑ꢁ  
퐼
= −ꢃ(훼ꢀꢊ + ꢂꢊ + 푏푧ꢊ)  
푑푡  
Subscript I here indicates the inner solution.  
푑푦  
푑ꢁ  
퐼
퐼
If we set ꢃ → 0 as expected in the perturbation theory, we will obtain  
=
= 0. Hence, yI and zI are  
푑ꢈ  
푑ꢈ  
approximately constant throughout the system, that is, yI = y(0) and zI = z(0).  
Subsequently, substitution of zI = z(0) into the first equation of system (7) leads to  
푑푥  
퐼
= 푧(0) − 푐1ꢀ3 − 푐2ꢀ2 − 휇ꢀꢊ .  
(8)  
ꢊ
ꢊ
푑푡  
B3  
Matching and Uniform Approximation  
The inner solution which provides a good approximation in the transient period, together with the outer solution  
which provides a good approximation in the post-transient period, comprise a total solution for the system. These  
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solutions have a common limit or overlap term, that is, where the outer solution begins to take over from inner  
solution. Hence matching is required here to get the uniform approximation.  
By applying the matching condition to the voltages across the capacitors C2 on yI and yo, we get  
( )  
lim ꢂ표(ꢄ) = lim ꢂꢊ 휏 = y(0).  
푡→0 ꢈ→∞  
While applying the matching condition to the intensity of the electrical current in the inductor L1 on zI and zo  
gives  
( )  
( )  
lim 푧표(ꢄ) = lim 푧ꢊ 휏 = z 0 .  
푡→0 ꢈ→∞  
On the other hand, imposing the matching requirement on the voltages across the capacitors C1 gives  
( )  
lim ꢀ표(ꢄ) = lim ꢀꢊ 휏 = ηc  
푡→0 ꢈ→∞  
Subsequently, we can have the uniform approximation x, y and z by adding the inner and outer solutions and  
subtracting their common limit, that is,  
푐푢 = 푐ꢊ + 푐표 − ηc  
( )  
ꢂ푢 = ꢂꢊ + ꢂ표 − ꢂ 0 = ꢂ표  
( )  
푧푢 = 푧ꢊ + 푧표 − 푧 0 = 푧표  
where these solutions well approximate the system dynamics throughout the whole problem domain.  
RESULT DISCUSSION  
We take the parameters 푐1 = 44 , 푐2 = 41 , 휇 = 2, 훽 = 1, 푎 = 0.7, 푏 = 0.24 and ꢃ = 0.05 for the simulation of  
3
2
Chua circuit using singular perturbation theory.  
Outer Solution  
By considering the initial conditions x(0) = 0, y(0) = 1, z(0) = 1 and set ꢃ = 0, we obtain the outer approximation  
by solving the algebraic equations (5) and differential equations (6). The simulation was done based on the  
coding written via Matlab with a step size of 0.01 for ꢄ ∈ [0. 4]. The results of the outer approximation are  
presented in Figure 2.  
Figure 2: Outer approximations of x, y and z  
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Inner Solution  
On the other hand, we solve the equation (8) in the boundary layer to get the inner approximation. The simulation  
was done based on the coding written via Matlab with a step size of 0.001 for ꢄ ∈ [0, 0.05], that is, we want to  
observe the rapid change in the boundary layer particularly. The result is represented pictorially in the Figure 3.  
We can see that the voltages across the capacitors C1 at time t = 0 is equal to its initial value.  
Figure 3: Inner approximation for x  
An additional remark here is that we can see that both the scaled system in outer and inner approximations (in  
subsection B1 and B2) are equivalent to the original system (1). We have the small parameter ꢃ appears in the  
푑푥  
1
휖
푑푦  
퐼
푂
(
)
equation  
=
푓 ꢀꢋ, ꢂꢋ, 푧ꢋ of the outer solution, whereas ꢃ appears in the equations  
= ꢃ푓 (ꢀꢊ, ꢂꢊ, 푧ꢊ)  
1
2
푑푡  
푑푡  
푑ꢁ  
퐼
(
)
and  
= ꢃ푓 ꢀꢊ, ꢂꢊ, 푧ꢊ . This enables the singular perturbation procedure to take place so that outer and inner  
3
푑푡  
approximations are obtained in different time scales.  
Uniform approximation  
Subsequently, we proceed to match the outer and inner solutions. More precisely, the voltages across the  
capacitors C2 on yI and yo is  
( )  
( )  
lim ꢂ표(ꢄ) = lim ꢂꢊ 휏 = y 0 = 1.  
푡→0 ꢈ→∞  
While the intensity of the electrical current in the inductor L1 on zI and zo are matched where  
( )  
( )  
lim 푧표(ꢄ) = lim 푧ꢊ 휏 = z 0 = 1.  
푡→0 ꢈ→∞  
On the other hand, the matching requirement is applied on the voltages across the capacitor C1 and thus its  
( )  
common limit is lim ꢀ표(ꢄ) = lim ꢀꢊ 휏 = 0.1696 for this case.  
푡→0 ꢈ→∞  
As a consequence, the uniform approximations are for ꢀ, ꢂ and 푧 are  
ꢀ푢 = ꢀꢊ + ꢀ표 − 0.1696, ꢂ푢 = ꢂ표 ,  
푧푢 = 푧표 .  
The uniform approximation for x, y and z are presented in Figure 4.  
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Figure 4: Uniform approximations of x, y and z  
Comparison with Actual Solution  
To justify the approximations in this project, we also solve the system of ODEs (1) numerically by using the  
Matlab function ode15s. The comparison of the uniform approximations and actual solution can be seen in the  
Figure 5(a), Figure 5(b) and Figure 5(c).  
By using the outer approximation, the reduced ODEs system (6) can be solved in lower computational cost.  
Through the results shown in the Figure 2(a), Figure 2(b) and Figure 2(c), we can observe that yo and zo well-  
approximate the y and z, but the approximation of xo is not valid in the boundary layer, that is, xo (t = 0) ≠ x(t =  
0).  
On the other hand, Figure 3 is the inner solutions for the voltages across the capacitors C1. This solutions satisfy  
the given initial condition but fail to provide a good approximation after the fast transient period. The matching  
applied to the outer and inner solutions has provided a better approximation, that is, the uniform approximation.  
This can be seen in Figure 4(a) and Figure 5(a).  
Note here the maximum of absolute error 푒̂, used in the computation is defined as  
{| ( )  
|}  
푒̂  
= max 휇 ꢄ − 휇푢(ꢄ)  
0≤푡≤ꢇ  
where 휇(ꢄ) represents the solution of full system at time t while 휇푢(ꢄ) is the uniform approximation at time t.  
The maximum of the absolute errors of ꢀ, ꢂ and 푧 are 0.145, 0.0013 and 0.0096 respectively.  
CONCLUSION  
In brief, this project is concerned with the application of the singular perturbation theory in reducing the size of  
dynamic model of the electronic circuits. We have applied the singular perturbation theory to reduce the size of  
dynamic model of the Chua circuit. The uniform approximation in our study provides a reasonably good  
approximation with a lower computational cost. The use of the approach discussed in this project can be extended  
to deal with higher dimensional problem in future, for example, for the VLSI circuits.  
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ACKNOWLEDGEMENT  
The author would like to express her thanks to Faculty of Electronics and Computer Technology and Engineering  
(FTKEK) at Universiti Teknikal Malaysia Melaka (UTeM) for their assistance in acquiring the essential  
information and resources for the successful completion of the research.  
Figure 5: Uniform approximation and numerical solution of system of equations (1) for the voltages across the  
capacitors C1, capacitors C2 and inductor L1 respectively, with parameters 푐 = 44 , 푐2 = 41 , 휇 = 2, 훽 = 1, 푎 =  
1
3
2
0.7, 푏 = 0.24 and initial conditions: x(0) = 0, y(0) = 1 and z(0) = 1.  
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