Chaos Engineering and Control in Mobile Robotics Applications

Salah Nasr

, Amine Abadi

, Kais Bouallegue

and Hassen Mekki

Networked Objects Control and Communication Systems-Laboratory, National Engineering School of Sousse,

University of Sousse, Tunisia

Department of Electrical Engineering, Higher Institute of Applied Sciences and Technology of Sousse, Tunisia

Keywords: Chaos Theory, Chaos Engineering, Chaotic Mobile Robot, Chaos Control, Robotic Applications.

Abstract: This article briefly summarizes the theory of chaos and its applications. Firstly, we begin by describing

chaos as an aperiodic bounded deterministic motion, which is sensitive to initial states and therefore

unpredictable after a certain time. Then, fundamental tools of the chaos theory, used for identifying and

quantifying chaotic dynamics, are shared. The paper covers a main numerical approach to identify chaos

such as the Lyapunov exponents. Many important applications of chaos in several areas such as chaos in

electrical and electronic engineering and chaos applications in robotics have been presented. An analysis of

the reviewed publications is presented and a brief survey is reported as well.

1 INTRODUCTION

During the 20th century, three great revolutions

occurred: quantum mechanics, relativity and chaos.

The theory of chaos, also called dynamical systems

theory, is the study of unstable aperiodic behavior in

deterministic dynamical systems, which show a

sensitive dependence on initial conditions

(Vaidyanathan, 2013). The sensitive dependence on

initial conditions implies that arbitrary initial

conditions follow trajectories that move away from

one another after a certain time (Moon, 2008), as

shown in figure 1. Due to determinism (Morrison,

2012), chaos is predictable for the short time; but it

is unpredictable in the long run due to sensitivity to

initial conditions. Chaos is characterized by a large

sensitive dependence to the initial state, by its

inability to predict future consequences, by the

Lyapunov exponent (Kuznetsov, 2016), by its fractal

dimension, and so on.

The nonlinear dynamics and chaos terms have

become known to most scientists and engineers over

the past few decades. Nonlinearities occur in

feedback processes (Gaponov-Grekhov and

Rabinovich, 2011), in systems containing interacting

subsystems, and in systems interacting with the

environment.

This scenario is qualitatively and quantitatively

distinct from the situations where the perturbations

develop linearly. Thanks to the availability of high-

speed computers and new analytical techniques, it

has become clear that the chaotic phenomenon is of

a universal nature and has transverse consequences

in various areas of human endeavour.

The devices of the fire fighting and floor

cleaning have been developed by exploiting

autonomous mobile robots as useful tools in

activities that put the integrity of humans in danger,

such as monitoring and exploring of terrains for

explosives or dangerous materials and such as

intrusion patrols at military installations. This has

driven to the development of intelligent robotic

systems (Martins-Filho and Macau, 2007).

Therefore, the unpredictability of a trajectory is also

a crucial factor for the mission success for such an

autonomous mobile robot. To meet this challenge,

Sekiguchi and Nakamura suggested a strategy in

2001 to solve the problem of path planning based on

chaotic systems (Nakamura and Sekiguchi, 2001).

Figure 1: Two trajectories that start close to each other but

diverge within a few tens of seconds (Moon, 2008).

364

Nasr, S., Abadi, A., Bouallegue, K. and Mekki, H.

Chaos Engineering and Control in Mobile Robotics Applications.

DOI: 10.5220/0006867103640371

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 2, pages 364-371

ISBN: 978-989-758-321-6

A key property of chaos is that simple dynamical

systems can often engender complex dynamics.

These systems can be implemented using simple

analogue hardware (Aihara, 2012).

In order to explore the applications of chaos in

engineering and robotics, this paper is organized as

follows. Section 2 gives an overview of the chaos

theory. In section 3, we review the research on

applications of chaotic dynamics in electrical and

electronic engineering. Chaos synthesis in robotics is

presented as an application of chaotic systems for

motion planning of autonomous mobile robots in

section 4. The conclusion is presented in section 5.

2 CHAOS THEORY:

AN OVERVIEW

The movements of several natural or engineering

systems can be governed by a set of equations

derived from natural laws such as the Newton's laws

or the Euler equation. The equations that describe a

dynamic system can be algebraic or differential. The

set of equations, mathematically defined as a

dynamic system, gives the temporal evolution of the

state of a system from the knowledge of its previous

history. Therefore, the state at any time can be

determined by the governing equations and the

initial states. In modern science, the term chaos is

used to describe a type of motion resulting from a

dynamic system that appears to be disordered and

extremely complex under detailed examination.

Complicacy and disorder are due to the reasons that

chaos is a recurrent aperiodic motion. Hence, chaos

can be defined as a bounded steady-state response

that is not an equilibrium state, a periodic motion, or

a quasi-periodic motion.

Chaotic movements are also characterized by

sensitivity to initial states; i.e, a simple variation in

the initial conditions can quickly produce enormous

differences in response. The long-term prediction of

chaos is impossible, due to such sensitivity. In other

words, chaos is unpredictable after some time

because a small difference in initial conditions

beyond their precision will result in a rapid growing

of the movement.

A dynamical system is called chaotic if it

satisfies the three properties: boundedness, infinite

recurrence, and sensitive dependence on initial

conditions (Azar and Vaidyanathan, 2015, Azar and

Vaidyanathan, 2014). The Chaos theory investigates

the qualitative and numerical study of unstable

aperiodic behaviour in deterministic nonlinear

dynamical systems.

In 1963, Lorenz (Lorenz, 1963) discovered a 3-D

chaotic system when he was studying a 3-D weather

model for atmospheric convection. After a decade,

Rössler (Rössler, 1976) discovered a 3-D chaotic

system, which was constructed during the study of a

chemical reaction. These classical chaotic systems

paved the way to the discovery of a lot of 3-D chao-

tic systems such as the Arneodo system (Arneodo et

al., 1981), the Chen system (Chen and Ueta, 1999),

the Lü-Chen system (Lü and Chen, 2002), the Tigan

system (Tigan and Opriş, 2008), etc.

Some other scientists have tried to propose new

dynamic systems. In (Leonov and Kuznetsov, 2013),

Leonov gave a survey on the relationship between

hidden oscillators and hidden chaotic attractors.

Snchez-Lpez et al. (Sánchez-López et al., 2010)

proposed a method to generate a multi-scroll chaotic

attractor based on Saturated Nonlinear Function

Series. In (Zhang and Tang, 2012), Zhang and Tang

introduced a new chaotic system with four

components that can generate chaos, hyperchaos,

and periodic and quasi-periodic behaviors. Sun et al.

(Sun et al., 2014) presented a finite-time

combination scheme of four chaotic systems and

solved the problem of the synchronization of two

systems. Bouallegue (Bouallegue, 2015b) found a

method to generate a new class of chaotic attractors

that possessed a multi-fractal scroll based on a

fractal process, as illustrated in figure 2. Another

work by Kais Bouallegue (Bouallegue, 2015a) and

Salah Nasr (Salah NASR, 2015) put forward a new

behavior of chaotic attractors with separated scrolls

using the combination between the fractal process

and the chaotic attractors as depicted in figure 3.

Figure 2: Chaotic attractor with fractal and multifractal

scrolls. (a) Multichaotic attractor with Lorenz system, (b)

multichaotic attractor with Chua system and (c)

multichaotic attractor with Rössler system (Bouallegue,

2015b).

Chaos Engineering and Control in Mobile Robotics Applications

365

Figure 3: Chaotic attractors with four behavior forms

(Salah NASR, 2015).

2.1 Identification of Chaos

The identification of chaos is the set of diagnostic

tests to determine if chaotic behavior occurs in a

specific system. To identify chaos, some numerical

characteristics associated with the motion of a

system can be used. These characteristics include the

Lyapunov exponents, power spectra and entropies.

2.2 Lyapunov Exponent

To estimate and compute the Lyapunov dimension,

Leonov (Kuznetsov et al., 2016, Kuznetsov, 2016)

proposed an analytical approach in 1991. A

Lyapunov exponent may be positive or negative. A

positive Lyapunov exponent implies the divergence

in the corresponding direction. A negative exponent

implies the constriction in the corresponding

direction. Hence, most dynamic systems can be

characterized by their Lyapunov exponents

(Kaygisiz et al., 2011). For a chaotic attractor, a two-

dimensional system must have a positive and a

negative Lyapunov exponents. If the system has two

zero Lyapunov exponents and two negative ones

then, the system is quasi-periodic. If the four-

dimensional system is hyperchaotic, it should have

two positives, a zero one and a negative one.

3 CHAOS ENGINEERING

APPLICATIONS

Over the last few decades, the terms chaos and

nonlinear dynamics are known to most engineers

and scientists. Nonlinearities occur in feedback

processes, in systems containing interacting

subsystems and in systems interacting with the

environment. It is striking to note that simple

devices, such as a double pendulum, and a very

complex event such as time follow the same

dynamics, which can only be predicted for short

time horizons (Kaygisiz et al., 2011). Thanks to the

existence of high-speed computers, new analytical

techniques and sophisticated experiments, it has

become clear that the chaotic phenomenon is of a

universal nature and has transverse consequences in

various fields of human endeavour.

Recently, chaos theory is found to have

important applications in several areas such as

biology (Garfinkel, 1992, May, 1976, Vaidyanathan,

2015a, Vaidyanathan, 2015b), memristors (Pham et

al., 2015, Volos et al., 2015a), electrical circuits

(Volos et al., 2015b), robotics (Nakamura and

Sekiguchi, 2001), etc.

A lot of practical applications of deterministic

chaos have been developed in various fields of

engineering and technology. Actually, studies of

nonlinear dynamics in engineering disciplines have

been steadily progressing over a half century.

Among these disciplines, we are going to deal with

electrical and electronic engineering, and

synchronization of chaotic systems.

3.1 Electrical and Electronic

Engineering

A lot Electrical and electronic circuits are typical

fields in which rich chaotic phenomena have been

reported. In 1927, Van der Mark and Van der Pol

heard noise corresponding to deterministic chaos in

an electrical system composed of a resistor, a neon

glow lamp, a variable condenser, and D.C. and A.C.

power sources (Van der Pol and Van der Mark,

1927). Since then, chaos has been detected both

experimentally and numerically in many electrical

and electronic circuits like the Chua's circuit, the

Shinriki's circuit (Chua et al., 1993, van Wyk and

Steeb, 1997) and the series RLC circuit with a

varactor diode (Testa et al., 1982). It should be noted

that both the Duffing equation and the Duffing-Van

Der Pol mixed type equation are also models of

electrical circuits with a nonlinear inductance and a

nonlinear resistance, respectively.

In order to create controlled chaotic outputs, the

work presented in (Hanias and Tombras, 2009,

Hanias et al., 2010) makes the study of simple

circuits which are composed of simple transistors

triggered externally be able to give significant

results for the possibility of creating its chaotic

outputs. These results are very important, especially

in telecommunication systems where the need for

safe signal transmission is imperative. In their work,

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366

the authors presented the chaotic behavior of three

forms of these circuits as well as corresponding

theoretical methodology. Next, in the study of

simple chaotic circuits, we consider the case of an

optoelectronic chaotic circuit which is based on an

optocoupler device and which can be used as a

controlled optoelectronic chaotic signal generator,

presented in (Hanias, 2010).

These studies have confirmed that chaos can be

actually and easily generated in nonlinear electrical

and electronic circuits.

3.2 Synchronization of Chaotic System

In parallel with the great advances in the chaos

theory, the prospects of utilizing chaos in various

applications, particularly in telecommunication,

have motivated researchers to study the possibility

of synchronizing chaos. The synchronization of non-

linear oscillators is a phenomenon which has

attracted the attention of researchers since the

discovery and description of this phenomenon by

Huygens in 1673, in an example of two coupled

mechanical systems. The phenomenon of

synchronization is manifested when two dynamic

systems evolve in an identically as a function of

time.

Since this innovative discovery, different

synchronization regimes have been distinguished

such as identical synchronization (Pecora and

Carroll, 1990) and generalized synchronization

(Rulkov et al., 1995). In (Jemaâ-Boujelben and Feki,

2016, Feki, 2004), the authors suggested a simple

Multi-input Multi-output (MIMO) adaptive control

based on a Sliding Mode Observer (SMO) to

synchronize chaos in the PMSM.

Similar to the integration of order chaotic

systems, the synchronization of fractional order

chaotic systems has interested several researchers

(Wang and Song, 2009, Tang and Fang, 2010). In

(Kiani-B et al., 2009), the synchronization of

fractional chaotic systems using the fractional

extended Kalman filter has were presented with an

application to secure communication. To

synchronize uncertain fractional order chaotic

systems, the authors used the adaptive fuzzy sliding

mode control in (Lin et al., 2011), and in

(Senejohnny and Delavari, 2012) the authors

employed a combination of a classical sliding

observer and an active observer. The generalized

synchronization was also addressed in the context of

fractional order chaotic systems.

4 CHAOTIC APPLICATIONS IN

ROBOTICS

We hope time is ripe for reviewing the application of

chaos in robotics. The community of robotics is

trying to emulate these natural behaviours by

investigating humanoids, bio-robots and the

biologically inspired systems such as swarms. These

systems confront problems like vibration, noise-

sensing and robot environment interactions leading

to chaos.

As a very interesting topic, Robotics and

particularly their applications have emerged in

various activities. In the military field, robotic

systems should have an interesting feature as target

identification and perception as well as positioning

robots on the ground. Moreover, the greatest

challenge for those successful robot missions is the

optimal path planning. Such nonlinearities have led

researchers to utilize chaotic trajectory planning

techniques for autonomous mobile robots to ensure a

rapid search of the whole workspace(Kaygisiz et al.,

2011) . The aim of the employment of chaotic

signals for autonomous mobile robots is to benefit

from coverage areas got through their movement

paths.

To meet this challenge Sekiguchi and Nakamura

suggested a strategy in 2001 to solve the problem of

path planning based on chaotic systems (Nakamura

and Sekiguchi, 2001). In that work, the chaotic

behavior of the Arnold dynamical system was

imparted to the mobile robot’s motion control.

Figure 4: The chaotic autonomous mobile robot (Volos et

al., 2013a).

Since then, a lot of relative researches have been

presented by many research teams because the

chaotic movement of the robot ensures the scanning

of the entire workspace without previous knowledge

of the terrain map. In (Volos et al., 2013a), a new

navigation strategy by designing a controller was

experimentally investigated, which ensured a chaotic

Chaos Engineering and Control in Mobile Robotics Applications

367

motion to an autonomous mobile robot wich is

presented in figure 4. The proposed controller

produced an unpredictable trajectory by imparting

the system’s chaotic behavior to the two independent

active wheels of the mobile robot.

In figure 5 a, the diagram shows the color scale

map of the terrain’s cells versus the times of visiting.

There are cells, wherein the robot has visited from 1

to up 18 times in the 24 minutes of operation. In

addition, the coverage rate versus the number of

visited cells, for the robot with the proposed chaotic

motion controller, is presented in figure 5 b.

Finally, according to the experimental results, the

high unpredictability of the robot’s trajectory, which

is very crucial in many tasks, is confirmed by

utilizing the suggested chaotic motion controller.

Furthermore, other two crucial tasks are succeeded

such as the complete and fast scanning of the

workspace.

(a)

(b)

Figure 5: (a) Color scale map of the terrain’s cells versus

the times of visiting, (b) Coverage rate versus number of

motion commands, for operation of 16 minutes. (Volos et

al., 2013a).

Similarly, a chaotic path planning generator for a

mobile robot was implemented by Volos et al. to

cover the overall workspace in an erratic and swift

manner. Therefore, the authors presented a Khepera,

which is popular in the robotics community,

integrating a behaviour-based control (Volos et al.,

2012a, Volos et al., 2012b). The chaotic generator

used three different chaotic systems producing a

double-scroll chaotic attractor.

Furthermore, Volos et al. put forward a motion

control strategy for humanoid and mobile robots

utilizing a chaotic truly random bit generator (Volos

et al., 2012d, Volos et al., 2012c). They implement-

ed an autonomous robot on an experimental platform

called the “Magician Chassis” using this generator.

This technique ensured highly unpredictable robot

trajectories that are random from an observer’s

viewpoint (Volos et al., 2013a, Volos et al., 2013b).

The numerical simulations demonstrated the

efficiency of this strategy and the statistical tests

also proved the randomness of the planned motions.

(a)

(b)

Figure 6: Intermediate iteration points distribution in the

phase space and their trajectories: (a) Iteration points; (b)

Iteration trajectories (Li et al., 2015).

Other proposals such as the studies of Caihong et al.

introduced a chaotic path planner based on a logistic

map. This deterministic and simple system was

characterized by a random behaviour, allowing large

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

368

workspace coverage (Li et al., 2013). Morever, they

suggested a fusion strategy to develop a chaotic path

planner for mobile robots, based on the standard

map (Li et al., 2015). Figure 6 analyzes the iteration

points distribution of the standard map in the phase

space and the adjacent qualitative and quantitative

trajectories. Using the original Standard map of the

chaotic state as a robot path planner generator to

perform the surveillance mission is a good choice

except for its large distance of the chaotic trajectories

between the iteration adjacent intermediate points.

To accomplish boundary surveillance missions

for mobile robots, Curiac and Volosencu proposed

an improved chaotic path planning technique

(Curiac and Volosencu, 2014). This study underpin-

ned the testing and design of a chaotic controller

incorporating a known chaotic equation. While

performing monitoring and research tasks, the

chaotic trajectories for autonomous robots had a

great enhancement over other methods. Security

patrol, fire fighting and cleaning were included in

the proposed applications of chaotic mobile robots.

As depicted in figure 7, the authors started with the

periodic motion on a closed contour of a reference

frame in which the Henon chaotic system would

evolve. They proved that the compound trajectories

obtained in the fixed frame were also chaotic. Based

on this result, they developed an original method to

create chaotic trajectories in the proximity of any

arbitrary boundary shape.

Figure 7: Chaotic robot path in the proximity of a closed

curve (two complete laps along the guiding line) (Curiac

and Volosencu, 2014).

These efforts show that the application of the chaotic

behaviour of dynamic systems for the planning of

mobile robot movements is a fascinating

interdisciplinary research field.

Unlike other path planning methods, chaotic path

planning does not require a map of the workspace

and it is more efficient than the random walk

algorithm. With various chaos equations, a robot

could exhibit a range of motion paths. The chaotic

controllers can be implemented by embedding

simple chaotic circuits into the robots. Chaotic

trajectories are generated using state variables of

dynamical systems, which are used as input for the

wheels of differential-drive robots.

5 CONCLUSIONS

This paper presents the concept of chaos, which

leads to several understandings of chaos. The theory

of chaos reveals our inability to make long-term

predictions of these deterministic dynamic systems.

Chaotic dynamics can be explained, measured and

categorized using this theory. The growth of research

on chaos is highlighted by the interdisciplinary nature

of the field. Due to several applications in electrical

appliances, the application of deterministic chaos has

attracted a lot of attention. The deterministic chaos

has been perceived as unpredictable and unstable, and

therefore worthless. Over the past decades, its

usefulness and application have been recognized.

Like other fields of science and technology, chaotic

dynamics have been discovered and implemented in

various fields of robotics.

In addition, some efforts for uncovering the

chaotic behaviour of various robots are presented.

This part provides basic knowledge about the

common methods and processes involved in finding

the evidence for the existence of chaos in robotic

motion, which can help in better applications of

chaos to real robots.

REFERENCES

Aihara, K. 2012. Chaos and its applications. Procedia

IUTAM, 5, 199-203.

Arneodo, A., Coullet, P. & Tresser, C. 1981. Possible new

strange attractors with spiral structure. Communica-

tions in Mathematical Physics, 79, 573-579.

Azar, A. T. & Vaidyanathan, S. 2014. Computational

intelligence applications in modeling and control,

Springer.

Azar, A. T. & Vaidyanathan, S. 2015. Chaos modeling

and control systems design, Springer.

Bouallegue, K. 2015a. Chaotic attractors with separated

scrolls. Chaos: An Interdisciplinary Journal of

Nonlinear Science, 25, 073108.

Bouallegue, K. 2015b. Gallery of chaotic attractors

generated by fractal network. International Journal of

Bifurcation and Chaos, 25, 1530002.

Chen, G. & Ueta, T. 1999. Yet another chaotic attractor.

Chaos Engineering and Control in Mobile Robotics Applications

369

International Journal of Bifurcation and chaos, 9,

1465-1466.

Chua, L. O., Wu, C. W., Huang, A. & Zhong, G.-Q. 1993.

A universal circuit for studying and generating chaos.

I. Routes to chaos. IEEE Transactions on Circuits and

Systems I: Fundamental Theory and Applications, 40,

732-744.

Curiac, D.-I. & Volosencu, C. 2014. A 2D chaotic path

planning for mobile robots accomplishing boundary

surveillance missions in adversarial conditions.

Communications in Nonlinear Science and Numerical

Simulation, 19, 3617-3627.

Feki, M. 2004. Synchronization of chaotic systems with

parametric uncertainties using sliding observers.

International Journal of Bifurcation and Chaos, 14,

2467-2475.

Gaponov-Grekhov, A. V. & Rabinovich, M. I. 2011.

Nonlinearities in Action: Oscillations Chaos Order

Fractals, Springer Publishing Company, Incorporated.

Garfinkel, A. 1992. Controlling cardiac chaos. Science.

Hanias, M., Giannis, I. & Tombras, G. 2010. Chaotic

operation by a single transistor circuit in the reverse

active region. Chaos: An Interdisciplinary Journal of

Nonlinear Science, 20, 013105.

Hanias, M. & Tombras, G. 2009. Time series cross

prediction in a single transistor chaotic circuit. Chaos,

Solitons & Fractals, 41, 1167-1173.

Hanias, P. M., Nistazakis, H. E., Tombras, G. S., 2010.

Study of an optoelectronic chaotic circuit.

International Interdisciplinary Chaos Symposium on

Chaos and Complex Systems, Istanbul, Turkey.

Jemaâ-Boujelben, S. B. & Feki, M. 2016. Synchronisation

of chaotic permanent magnet synchronous motors via

adaptive control based on sliding mode observer.

International Journal of Automation and Control, 10,

417-435.

Kaygisiz, B. H., Karahan, M., Erkmen, A. M. & Erkmen,

I. 2011. Robotic approaches at the crossroads of

Chaos, fractals and percolation theory. Applications of

Chaos and Nonlinear Dynamics in Engineering-Vol.

1. Springer.

Kiani-B, A., Fallahi, K., Pariz, N. & Leung, H. 2009. A

chaotic secure communication scheme using fractional

chaotic systems based on an extended fractional

Kalman filter. Communications in Nonlinear Science

and Numerical Simulation, 14, 863-879.

Kuznetsov, N. 2016. The Lyapunov dimension and its

estimation via the Leonov method. Physics Letters A,

380, 2142-2149.

KUZNETSOV, N., ALEXEEVA, T. & LEONOV, G.

2016. Invariance of Lyapunov exponents and

Lyapunov dimension for regular and irregular

linearizations. Nonlinear Dynamics, 85, 195-201.

Leonov, G. A. & Kuznetsov, N. V. 2013. Hidden

attractors in dynamical systems. From hidden

oscillations in Hilbert–Kolmogorov, Aizerman, and

Kalman problems to hidden chaotic attractor in Chua

circuits. International Journal of Bifurcation and

Chaos, 23, 1330002.

Li, C., Song, Y., Wang, F., Liang, Z. & Zhu, B. 2015.

Chaotic path planner of autonomous mobile robots

based on the standard map for surveillance missions.

Mathematical Problems in Engineering, 2015.

Li, C., Wang, F., Zhao, L., Li, Y. & Song, Y. 2013. An

improved chaotic motion path planner for autonomous

mobile robots based on a logistic map. International

Journal of Advanced Robotic Systems, 10, 273.

Lin, T.-C., Lee, T.-Y. & Balas, V. E. 2011. Adaptive

fuzzy sliding mode control for synchronization of

uncertain fractional order chaotic systems. Chaos,

Solitons & Fractals, 44, 791-801.

Lorenz, E. N. 1963. Deterministic nonperiodic flow.

Journal of the atmospheric sciences, 20, 130-141.

Lü, J. & Chen, G. 2002. A new chaotic attractor coined.

International Journal of Bifurcation and chaos, 12,

659-661.

Martins-Filho, L. S. & Macau, E. E. 2007. Trajectory

planning for surveillance missions of mobile robots.

Autonomous Robots and Agents. Springer.

May, R. M. 1976. Simple mathematical models with very

complicated dynamics. Nature, 261, 459-467.

Moon, F. C. Chaotic and Fractal Dynamics. Introduction

for Applied Scientists and Engineers: John Wiley &

Sons.

Morrison, F. 2012. The art of modeling dynamic systems:

forecasting for chaos, randomness and determinism,

Courier Corporation.

Nakamura, Y. & Sekiguchi, A. 2001. The chaotic mobile

robot. IEEE Transactions on Robotics and

Automation, 17, 898-904.

Pecora, L. M. & Carroll, T. L. 1990. Synchronization in

chaotic systems. Physical review letters, 64, 821.

Pham, V., Volos, C. K., Vaidyanathan, S., Le, T. & Vu, V.

2015. A memristor-based hyperchaotic system with

hidden attractors: dynamics, synchronization and

circuital emulating. Journal of Engineering Science

and Technology Review, 8, 205-214.

Rössler, O. E. 1976. An equation for continuous chaos.

Physics Letters A, 57, 397-398.

Rulkov, N. F., Sushchik, M. M., Tsimring, L. S. &

Abarbanel, H. D. 1995. Generalized synchronization

of chaos in directionally coupled chaotic systems.

Physical Review E, 51, 980.

Salah Nasr, K. B., Hassen Mekki 2015. Hyperchaos Set

By Fractal Processes System 8th International

CHAOS Conference Proceedings Henri Poincare

Institute, Paris, France

Sánchez-López, C., Trejo-Guerra, R., Munoz-Pacheco, J.

& Tlelo-Cuautle, E. 2010. N-scroll chaotic attractors

from saturated function series employing CCII+ s.

Nonlinear Dynamics, 61, 331-341.

Senejohnny, D. M. & Delavari, H. 2012. Active sliding

observer scheme based fractional chaos synchroniza-

tion. Communications in Nonlinear Science and

Numerical Simulation, 17, 4373-4383.

Sun, J., Shen, Y., Wang, X. & Chen, J. 2014. Finite-time

combination-combination synchronization of four

different chaotic systems with unknown parameters

via sliding mode control. Nonlinear Dynamics, 76,

383-397.

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

370

Tang, Y. & Fang, J.-A. 2010. Synchronization of N-

coupled fractional-order chaotic systems with ring

connection. Communications in Nonlinear Science

and Numerical Simulation, 15, 401-412.

Testa, J., Pérez, J. & Jeffries, C. 1982. Evidence for

universal chaotic behavior of a driven nonlinear

oscillator. Physical Review Letters, 48, 714.

Tigan, G. & Opriş, D. 2008. Analysis of a 3D chaotic

system. Chaos, Solitons & Fractals, 36, 1315-1319.

Vaidyanathan, S. 2013. Analysis and adaptive

synchronization of two novel chaotic systems with

hyperbolic sinusoidal and cosinusoidal nonlinearity

and unknown parameters. Journal of Engineering

Science and Technology Review, 6, 53-65.

Vaidyanathan, S. 2015a. Adaptive backstepping control of

enzymes-substrates system with ferroelectric

behaviour in brain waves. International Journal of

PharmTech Research, 8, 256-261.

Vaidyanathan, S. 2015b. Adaptive biological control of

generalized Lotka-Volterra three-species biological

system. Int J PharmTech Res, 8, 622-631.

Van Der Pol, B. & Van Der Mark, J. 1927. Frequency

demultiplication. Nature, 120, 363-364.

Van Wyk, M. & Steeb, W. 1997. Chaos in Electronics,

series Mathematical Modelling: Theory and

Applications, Vol. 2. Kluwer Academic Publishers,

Dordrecht, The Netherlands.

Volos, C. K., Bardis, N., Kyprianidis, I. M. & Stouboulos,

I. N. 2012a. Implementation of mobile robot by using

double-scroll chaotic attractors. Recent Researches in

Applications of Electrical and Computer Engineering,

119-124.

Volos, C. K., Bardis, N., Kyprianidis, I. M. & Stouboulos,

I. N. 2012b. Motion control of a mobile robot based on

double-scroll chaotic circuits. WSEAS Trans. Systems,

11, 479-488.

Volos, C. K., Doukas, N., Kyprianidis, I., Stouboulos, I. &

Kostis, T. Chaotic autonomous mobile robot for

military missions. Proceedings of the 17th

International Conference on Communications, 2013a.

Volos, C. K., Kyprianidis, I. & Stouboulos, I. 2012c.

Motion control of robots using a chaotic truly random

bits generator. Journal of Engineering Science and

Technology Review, 5, 6-11.

Volos, C. K., Kyprianidis, I., Stouboulos, I., Tlelo-

Cuautle, E. & Vaidyanathan, S. 2015a. Memristor: A

new concept in synchronization of coupled

neuromorphic circuits. Journal of Engineering Science

and Technology Review, 8, 157-173.

Volos, C. K., Kyprianidis, I. M. & Stouboulos, I. N.

2012d. A chaotic path planning generator for

autonomous mobile robots. Robotics and Autonomous

Systems, 60, 651-656.

Volos, C. K., Kyprianidis, I. M. & Stouboulos, I. N.

2013b. Experimental investigation on coverage

performance of a chaotic autonomous mobile robot.

Robotics and Autonomous Systems, 61, 1314-1322.

Volos, C. K., Pham, V.-T., Vaidyanathan, S., Kyprianidis,

I. & Stouboulos, I. 2015b. Synchronization

Phenomena in Coupled Colpitts Circuits. Journal of

Engineering Science & Technology Review, 8.

Wang, X.-Y. & Song, J.-M. 2009. Synchronization of the

fractional order hyperchaos Lorenz systems with

activation feedback control. Communications in

Nonlinear Science and Numerical Simulation, 14,

3351-3357.

Zhang, J. & Tang, W. 2012. A novel bounded 4D chaotic

system. Nonlinear Dynamics, 67, 2455-2465.

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