Exploring Non-linear Dynamics: Constructing the Bifurcation Diagram

of a Damped Driven Pendulum using Python

JOAN JANI

Department of Engineering Physics,

Polytechnic University of Tirana,

Rr. S. Delvina, 1001,

ALBANIA

Abstract: In this paper, we present a detailed analysis and construction of the bifurcation diagram for the

damped-driven pendulum system. The bifurcation diagram in general presents the qualitative changes of the

steady-state behavior for the pendulum. For this purpose, we implement the use of the Python programming

language with the inclusion of scientific libraries. This nonlinear dynamical system is an example of a system that

exhibits a chaotic regime, which is the sensitivity of its behavior to the initial conditions and the parameters of the

system. We investigate the response of the system to a range of drive strengths γapplied. By changing the driving

strength, the system reveals patterns of periodicity, quasi-periodic, and chaotic regimes. The critical values where

it goes from regular motion to chaotic one are highlighted, offering further understanding of the mechanisms of the

transitions. This study presents the use of the Python programming language for the modeling and visualization

of non-dynamic systems and contributes to a deeper understanding of nonlinear oscillator dynamics.

Key-Words: - Bifurcation Diagram, Damped Driven Pendulum, Non-linear Dynamics, Python Programming,

Numerical Methods, Chaotic Systems, Periodicity.

Received: March 19, 2024. Revised: August 11, 2024. Accepted: September 12, 2024. Published: October 24, 2024.

1 Introduction

In complex systems, the state of apparently

stochastic and unpredictable behavior derived from a

deterministic description of them is called chaos. This

concept defines the limits of order and predictability,

[1]. Its presence is ubiquitous, from turbulence in

weather systems to the behavior of electronic circuits,

[2]. This behavior generally comes from nonlinear

systems under certain conditions; this means that

not all nonlinear systems are chaotic, [3]. During a

chaotic regime, the behavior of the system becomes

unpredictable, giving the impression of stochastic

and random operation, [4].

The bifurcation diagram is a powerful tool for

understanding the evolution of a complex system by

changing a parameter of them, [5]. The bifurcation

diagram plots the possible steady state values of the

system against a varying parameter, [6]. This diagram

starts with a stable state, and as the parameters

change, these states branch into multiple branches;

this is an indication of different behaviors. The points

where the graph is splitting are known as bifurcation

points, [7].

The logistic map is an example of a bifurcation

diagram in which the parameter of the growth rate

increases. In this diagram, the population can first

stabilize in a value, then oscillate between two values

and eventually become chaotic, [8]. The stability

and periodicity within the system reveal the intricate

transition to chaos. This reveals the delicate interplay

between order and disorder in natural phenomena, [9].

The use of advanced computational tools for

modeling, data analysis, and visualization of the

results in the fields of dynamical systems creates new

opportunities for research, [10]. The widespread use

of Python as a tool for studying dynamical systems

has been well established during recent years. There

are numerous examples presented in literature for the

application of Python in studying dynamical systems,

[11], [12], [13]. The potential use of chaotic systems

for cryptography purposes and their use in secure

communication systems in general has increased the

interest in their study, [14].

In this paper, we investigate the dynamics of the

non-linear dumped-driven pendulum and explore the

construction of bifurcation diagrams using Python

programming. Several analytical solutions exist in

the literature for the damped pendulum, but when

an external force is applied, the solution is typically

only approximated numerically, [15], [16]. The

proposed method is efficient and easy to implement in

other chaotic systems. Another advantage is that our

proposal is based on free source code, available in the

public domain, and the program used and described

in this paper.

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2 Problem Formulation

We present a new approach to constructing the

bifurcation diagram in a dynamic system. We have

chosen Python as the programming language for this

purpose. We will use it to model the system, simulate

it, and present the results.

A nonlinear damped pendulum is an extension of

the simple pendulum that incorporates both damping

effects and nonlinearity in its restoring force. This

system is governed by a differential equation that

accounts for the pendulum’s angle, damping, and

nonlinear restoring forces, [6]. Such pendulums are

found in various natural and engineered systems,

and their study provides insight into the dynamics

of oscillatory systems with energy dissipation and

nonlinearities.

Fig. 1: A simple pendulum where three forces are

applied, the weight, the resistive force and the driven

force, [6]

The equation describing the motion of the

pendulum can be derived from the equation of a rigid

object under a net torque I¨

θ= Γ, where Idenotes the

moment of inertia of the object in our case I=mL2,

and Γis the net torque about the pivot point. Figure

(1) shows three forces applied on the object: the

weight of the object mg applies a torque −mgL sin θ,

the resistive force is proportional to the speed of the

object and has a magnitude bv, hence exerts a torque

−Lbv =−bL2˙

θand the driving force F(t)applying

a torque LF (t). Thus, the equation of motion:

mL2¨

θ=−mgL sin θ−bL2˙

θ+LF (t)(1)

The drive external force F(t)is sinusoidal given by:

F(t) = Acos(ωDt)

where Ais the amplitude of the driving force (in

Newtons) and ωDis the drive frequency. Rearranging

the equation (1) we have and substituting the

expression for the external force we have:

θ+g

Lsin θ+b

m˙

θ=A

mL cos ωDt(2)

By substituting the coefficient b/m= 2βwhere β

is called the damping constant, g/L=ω2

owhere

ωodenotes the natural frequency of the pendulum

and A

mL =γω2

owhere γdenotes the drive strength,

we have the differential equation of the motion for a

driven damped pendulum:

θ+ω2

osin θ+ 2β˙

θ=γω2

ocos ωt (3)

When the dimensionless parameter γis less than

1, it indicates that the driving force is weaker than the

pendulum’s weight. In this case, the motion induced

by the driving force is relatively small compared

to the overall dynamics governed by gravity and

damping, [6]. We investigate the motion of the

pendulum by varying the parameter γacross a range

of values.

3 Solution

We will use numerical methods to solve the equation

presented in (3). Numerical methods have been made

easier through their implementation in packages such

as Scipy in the Python programming language, [17].

The results from the simulation will be presented

using the matplotlib package, [18]. These tools are

easy to use and do not require a lot of computing

power. The parameters of the system are chosen

ωD= 2π,ωo= 1.5ωand β=ωo/4.

The second-order differential equation (3) is

rewritten as a system of two second-order equations

as follows:

dω

dt =−ω2

osin θ−2β˙

θ+γω2

ocos ωDt

dθ

dt =ω

(4)

The system of differential equations can be solved

through the function odeint, which is in the scipy

package of the Python ecosystem. The function

odeint is using lsoda from the FORTRAN library

odepack. For solving the equations we use γ= 0.9,

we use the initial conditions ˙

θ= 0 and several

values for the initial angle listed bellow, θ0=

[−2.5,−1.5,−0.5,0.5,0,1.5,2.5]. The solutions for

every initial angle value are plotted together and

presented in Fig. 2.

The first we notice in Fig. 2 is that the system,

after a transition state which depends on the initial

conditions, ends up in a steady state which does not

depend on the initial conditions. For a linear damped

oscillator subjected to a sinusoidal driving force,

there is a unique attractor that the system’s motion

will approach regardless of the initial conditions.

Moreover, the motion of the attractor is sinusoidal

and has a frequency that precisely corresponds to the

drive frequency. As can be seen here, the period of

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Fig. 2: The angle θof the pendulum as a

function of time for various initial conditions θ0=

[−2.5,−1.5,−0.5,0.5,0,1.5,2.5] when γ= 0.09

the motion is T= 1s, which is equal to the period

of the external force. So, in the linear regime, the

system obeys the frequency of the external force.

This behaviour is thoroughly analyzed in, [19]. This

will continue to happen as the value for γremains

sufficiently small. For some value of drive strength

γ, the attractor becomes unstable, and the system’s

behaviour becomes more complex.

Fig. 3: The angle θof the pendulum with time for two

different initial conditions with θ= 0 and θ= 0.001,

when drive strength = 1.5.

In Fig. 3, the angle θis presented as a function of

time when the value of = 1.503, two different initial

conditions θ= 0 and θ= 0.001. As can be seen from

the graph, at the beginning, the system’s behaviour

is the same for both initial conditions, but then their

difference begins to increase exponentially. The

Lyapunov exponent measures the rate of separation,

[20]. For a nonlinear oscillator, different initial

conditions can result in distinct attractors. Here, the

behaviour is not periodic as in Fig. 2, and the slight

difference in initial conditions leads the system to

different trajectories.

Οne way to show the correlation between the

parameter γand the transition of the system to chaos

is the bifurcation diagram, [6]. The construction of

this diagram is presented in detail in the following

paragraph.

4 Bifurcation

A bifurcation diagram presents the evolution of the

motion of θ(t)as the driving strength γchanges.

Our chosen values in this study are 1.03 ≤γ≤

2.15. A large number of values of γevenly

spaced, by an interval of ∆γ= 0.0001 could be

taken using the numpy function arange as follows:

numpy.arange(1.03,2.15,1e-5). By executing

this command, we get a matrix of 112000 elements

for the driving strength γ. Continuing our analysis,

we are solving the equations describing the system’s

behaviour (4) for each value of γpresented in

the matrix, with the same parameters and initial

conditions given in the previous paragraph.

The time for simulation is from 0 to 500 seconds;

the initial conditions are θ=−π/2 and ˙

θ= 0.

We choose the last 100 seconds to sample the angle

θthat the oscillator is at. Sampling is done at the

same frequency as the frequency of the external force

ωD= 2π, so Ts= 1s. The calculated values of θ

for each of the values of γare stored in the matrix

biff, and are presented in Fig. 4. The time needed

to run the code is approximately 5 hours on a laptop.

The code is publicly available and can be found here,

[21].

When the system has periodic behaviour,

sampling at times equal to its period will give

us the same value. So, only one value will appear

for a given value of γin the bifurcation diagram.

This occurs for small values of γfrom γ= 1.03

to γ= 1.06631. After the value γ= 1.06631, the

system displays two points for each value of γ. This

is referred to in the literature as bifurcation, and the

system has doubled its period. The period doubles

again for γ= 1.07951, where we can get four

different points for θ. The next bifurcation occurs

when γ= 1.08268. In more detail, our plot shows

the transition to chaos than the work done before,

[22].

For constructing the bifurcation diagram, we do

not require the pendulum’s motion to be limited in

space. We aim to study when the system goes

into chaotic behaviour for a more extensive range of

values −π≤θ≤πof the parameter g. We believe

that changing the angle’s values affects the emergence

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Fig. 4: Bifurcation diagram for 1.06 ≤γ≤1.09. The

period-doubling occurs for value γequal to 1.06631,

1.07935, 1.08207 and 1.08268.

Fig. 5: Bifurcation diagram for 1.03 ≤γ≤1.53.

of the system’s behaviour and, in general, its chaotic

operation.

In Fig. 5, the bifurcation diagram is present for

a wide range of values of the parameter g where

1.03 ≤γ≤1.53. For 1.26 ≤γ≤1.45 we

have a long stretch of period 1, followed by another

period-doubling cascade and the final section which

is primarily chaotic. What you notice is that in this

diagram, the dynamics of the system are the same

compared to diagram 12.18 form, [6].

In Fig. 6, after the chaotic section 1.495 ≤γ≤

1.55, the system exhibits periodic behaviour for the

values 1.54 ≤γ≤1.68.

The Fig. 7 presents the bifurcation diagram

when drive strength 1.65 ≤γ≤2.15. Here,

the chaotic behaviour is present for values 1.68 ≤

γ≤1.815. The system exhibits periodic behaviour

Fig. 6: Bifurcation diagram for 1.4≤γ≤1.85.

Fig. 7: Bifurcation diagram for 1.65 ≤γ≤2.15.

between 1.815 ≤γ≤1.865. The transition to

chaos is done with a period-doubling cascade, as seen

in Fig. 4. The chaotic regime is present for values

1.865 ≤γ≤2.125. After that, the system became

again periodic.

5 Conclusion

In this study, we have created the bifurcation diagram

for the damped-driven pendulum, focusing on the

transition from periodic to chaotic behavior. The

bifurcation diagram was contracted over extended

time periods. The results give a detailed bifurcation

diagram, where the complex behavior of the system

changes between periodic and chaotic regimes as the

drive strength parameter, γ, is varied.

When the drive strength value γis low, the system

shows periodic behavior that is characterized by a

single attractor that is not affected by the initial

conditions of the system. As γincreases, a series

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of period-doubling bifurcations are observed, which

are leading to the chaotic behavior. The transitions

from periodic to chaotic regimes are presented in the

bifurcation diagram, with the values of drive strength

for which this occurs. For values of γsmaller than

1.06631, the system is periodic. The first period

doubling occurs when γ= 1.07935 and the next at

γ= 1.0820. For values of γ > 1.08268, the system

enters into chaotic behavior.

The investigation of chaos generated by a

damped-driven pendulum could be considered a

case study with implications in various fields, like

chaos-based cryptography for applications in secure

communications. The investigation of conditions

where chaos emerges in this system gives us potential

for controlling chaotic systems through adjusting the

parameters precisely.

The complex dynamical systems could be

effectively analyzed with the Python programming

language and libraries like SciPy for scientific

computation and Matplotlib for visualization of

results. Our approach facilitated the exploration

of the pendulum’s dynamics and underscored the

usability of these methods and in other systems. The

code is available publicly available, and it can be

found on github, [21].

Overall, in this paper we have presented a

methodology for studying the transitions of the

system from periodic to chaotic behavior of a

nonlinear system using the bifurcation diagram.

Future work will extend this analysis to explore the

effects of additional parameters, such as changing

the damping parameter. Similar approaches could be

applied in other nonlinear dynamic systems.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from formulating the problem

to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflicts of Interest

The authors have no conflicts of interest to

declare that are relevant to the content of this

article.

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