Theoretical and numerical modelling of chaotic electrostatic ion cyclotron (EIC)oscillations by Jerk equationA. M. Wharton, Pankaj Kumar Shaw, M. S. Janaki, Awadhesh Prasad, and A. N. Sekar Iyengar Citation: Physics of Plasmas (1994-present) 21, 022311 (2014); doi: 10.1063/1.4865823 View online: http://dx.doi.org/10.1063/1.4865823 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/21/2?ver=pdfcov Published by the AIP Publishing

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Theoretical and numerical modelling of chaotic electrostatic ioncyclotron (EIC) oscillations by Jerk equation

A. M. Wharton,1,a) Pankaj Kumar Shaw,1 M. S. Janaki,1 Awadhesh Prasad,2

and A. N. Sekar Iyengar11Saha Institute of Nuclear Physics, Plasma Physics Division, 1/AF, Bidhannagar, Kolkata 700064, India2Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India

(Received 15 October 2013; accepted 3 February 2014; published online 25 February 2014)

In the last few years, third order explicit autonomous differential equations, known as jerk

equations, have generated great interest as they show features of regular and chaotic motion. In this

paper, we have modelled chaotic electrostatic ion cyclotron oscillations using a third order nonlinear

ordinary differential equation (ODE) and investigated its nonlinear dynamical properties. The

nonlinear ODE has been derived for a plasma system from a two fluid model in the presence of a

source term, under the influence of an external magnetic field, which is perpendicular to the

direction of the wave vector. It is seen that the equation does not require an external forcing term to

obtain chaotic behaviour. The stability of the solutions of the equation has been investigated

analytically as well as numerically, and the bifurcation diagram obtained shows a number of

interesting phenomena for various regimes of parameters. The coexisting attractors as well as their

corresponding basins are shown and the phase space portraits at different conditions are obtained

numerically and shown here. The results obtained here are in agreement with preliminary

experiments conducted for a similar configuration of a plasma system. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4865823]

I. INTRODUCTION

The study of nonlinear phenomena is an area of much

interest due to its usefulness in a number of diverse areas1,2

including plasma physics. Plasma, being a nonlinear and

complex system, is capable of sustaining a wide spectrum of

oscillations and instabilities depending on the configuration

and hence, its detailed investigation using nonlinear techni-

ques, will be very important to explain the various types of

observations, from space plasma3,4 to laboratory plasma,5

and also industrial plasmas.6,7 Conventional investigation

techniques such as dispersion analysis have been used to

unravel several underlying phenomena but nonlinear dynam-

ics as a tool has helped in understanding observations from

plasma systems better.8–12 Moreover, nonlinear dynamics

along with chaos theory involves a more detailed investiga-

tion of the transitions from quiescence to turbulence and vice

versa through the study of different bifurcation analysis such

as saddle node, transcritical, hopf, etc.13

A typical unmagnetised glow discharge plasma exhibits

a variety of oscillations ranging from chaotic to relaxation

type periodic oscillations5 and the usual trend to explain

most of the laboratory and space plasma experimental obser-

vations have been to deploy the Van der Pol14,15 or the

Duffing oscillator like equations due to an external forcing

term. Keen and Fletcher16–18 had obtained a second order

nonlinear ordinary differential equation (ODE) and exten-

sively discussed its dynamics due to external forcing. Kadji

et al.,19 using a similar approach, obtained an anharmonic

oscillator equation with which they investigated its attractor

dynamics from chaos point of view, for different values of

the external forcing amplitude. Chaturvedi20 studied

obliquely travelling Electrostatic ion-cyclotron (EIC) oscilla-

tions in a collisionless magnetoplasma and obtained special

steady state finite amplitude non sinusoidal saw tooth like

solutions. There has so far not been any study describing

chaotic EIC oscillations through a nonlinear ODE and the

same has been attempted for the first time here. Horton

et al.21 have proposed the “Horton-Weigel-Sprott” model to

explain the magnetosphere-ionosphere coupling through an

autonomous third order nonlinear differential equation. It is

commonly observed in many plasma experiments that there

is chaos without external forcing and hence, in this paper, we

attempt to explain the observed behaviour.

In the last few decades, the importance of jerk equations

in modelling the nonlinear behaviour of physical systems has

increased making this research area interesting.21–28 The

study of nonlinear phenomena obtained from jerk equations

is very interesting from mathematical point of view as well.

Despite the extensive work done on studying and classifying

nonlinear systems in the last few decades, there still remain a

number of unresolved questions, one of these being whether

there are more types of bifurcations and if so then under

what conditions do they appear. Jerk equations show a num-

ber of interesting nonlinear and chaotic phenomena, many of

which have not seen or studied before.

We had earlier obtained a third order nonlinear ODE,29

for chaotic electrostatic ion cyclotron oscillations, which

happen to be the dominant modes in a magnetised plasma,

starting from the fluid equations similar to Kadji et al.19 The

interesting aspect of this jerk equation is that it does not

require an external forcing term to obtain chaotic behaviour

as needed in most of the second order nonlinear ordinary

differential equations studied earlier. In this work, we havea)Electronic mail: alpha.wharton@saha.ac.in

1070-664X/2014/21(2)/022311/6/$30.00 VC 2014 AIP Publishing LLC21, 022311-1

PHYSICS OF PLASMAS 21, 022311 (2014)

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carried out detailed investigations on the nonlinear dynami-

cal properties of the equation obtained for a chosen set of

values representing physical parameters. The rest of this pa-

per is structured as follows. In Sec. II, we have provided

some details of the derivation of the jerk equation, and in

Sec. III, we have carried out detailed analysis of the equa-

tion from the nonlinear dynamics point of view for a fixed

value of control parameters which represent various physi-

cal quantities such as the external magnetic field. In this sec-

tion, we also present the bifurcation plots, obtained

numerically. It is seen that the results obtained numerically

are in agreement with those obtained analytically.

Section IV describes the stability analysis of the system

along with the different types of bifurcations. The coexisting

attractors and their corresponding basins are shown and dis-

cussed. Phase space plots for various values of the control

parameter are shown and the mathematical phenomena of

crisis as observed here in a physical system are discussed. It

is observed that the analytical and numerical results

obtained here are in agreement with preliminary experi-

ments conducted on a system with similar configuration.30

Section V gives a summary of the work done along with its

implications. We conclude by discussing the scope of this

work and our future plans.

II. MODEL AND EQUATION OF MOTION

EIC oscillations were first observed in laboratory experi-

ments by D’Angelo and Motley31 (1962). EIC waves are im-

portant in various natural systems such as auroral plasma

physics because EIC waves accelerate the ions in the direc-

tion perpendicular to the earth’s magnetic field, as well as in

laboratory systems.32

Here, we study ion cyclotron waves propagating in a

direction perpendicular to the magnetic field B0. In order to

explore the nonlinear dynamics of electrostatic ion cyclotron

waves propagating in a direction perpendicular to the mag-

netic field, we consider the plasma to be composed of two

interpenetrating fluids, similar to Kadji et al.19 We have

taken the source term of the form19 S ¼ �kn21 � �n3

1, which

represents nonlinear recombination instabilities.

In derivation of the equation, we have assumed the

Boltzmann relation for electrons and since we are dealing

with electrostatic oscillations, k�E¼ 0. As far as the ion

motion is considered,15–19 spatial variations are taken to be

of the form exp(ikx), where k is the wave number in the

x-direction; and the wave electric field is taken as E¼E1ex.

For the electrons, k is almost but not exactly perpendicular to

B, so that the electrons can move along the magnetic field to

enable Debye shielding, and also allows the use of the

Boltzmann relation for the electrons.

Since small amplitude waves are investigated, we use

the linearization technique in space. The potential / is

assumed to be 0 in the steady state. All unperturbed variables

are constant in space and time and the plasma approximation

leads to n1i¼ n1e¼ n1. In the nonlinear analysis, the space

part of the solution is assumed to be the same as in the linear

analysis, while the temporal part is governed by an appropri-

ate nonlinear equation.

Using the above considerations, the equation in the form

given below was derived by us,29 where x represents any

fluctuating quantity like the plasma density or the potential.

x000 þ ðaþ bxþ cx2Þx00 þ ðbþ 2cxÞx02

þ 1þ a2

4þ abxþ acx2

� �x0 þ d � a

2xþ b

2x2 þ c

3x3

� �

þ a

2xþ a2b

8x2 þ a2c

12x3

� �¼ 0: (1)

Equation (1) belongs to a class of third order nonlinear

ODE also called jerk equations22 which have a variety of

applications in different physical systems. They have been

studied extensively from academic point of view,33,34

but Horton et al.21 have obtained a jerk equation for

magnetosphere-ionosphere coupling. The second term is a

second order derivative, which corresponds to the dissipative

term. The third term corresponds to the square of the first

order derivative and the fourth term corresponds to the first

order derivative. The fifth and the sixth terms are the terms

corresponding to the restoring force. They could have been

combined into one but since the fifth term has the control pa-

rameter d, representing applied magnetic field, we have kept

it separately.

According to Lashinsky,35 an equation like Eq. (1) can

be used for description of plasma instabilities, if it satisfies

two conditions:

(1) The instability arises spontaneously.

(2) The instability must reach steady-state finite amplitude

and remains constant.

These properties are exactly those desired of plasma

instabilities. In Sec. III, we analyse the derived jerk

equation.

III. ANALYSIS OF JERK EQUATION

One reason to study the above obtained equation,

Eq. (1), is to look into the behavior of chaotic EIC oscilla-

tions. Another aspect is to study the equation from an

entirely nonlinear dynamical point of view and look into

interesting phenomena which show up. The third order jerk

equation can be rewritten as three first order ODE’s as

dx

dt¼ y;

dy

dt¼ z;

dz

dt¼ �ðaþ bxþ cx2Þz� ðbþ 2cxÞy2

� 1þ a2

4þ abxþ acx2

� �y� d � a

2xþ b

2x2 þ c

3x3

� �

� a

2xþ a2b

8x2 þ a2c

12x3

� �: (2)

In this paper, we fixed the parameters values: a¼ 0.5,

b¼�0.08, c¼ 0.002, which correspond to observations sim-

ilar to those obtained experimentally. It is also possible to

022311-2 Wharton et al. Phys. Plasmas 21, 022311 (2014)

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use this model as a starting point for other physical systems,

where such phenomena as described here are observed

experimentally as done by Horton et al.21

The possible fixed points of this system are (x01,2,3,

y0¼ 0, z0¼ 0), where trivial one is x01¼ 0, while nontrivial

x02 and x03 are the solutions of

a2

12þ d

3

� �cx2 þ a2

8þ d

2

� �bxþ ð1� dÞ a

2¼ 0: (3)

The two possible nontrivial solutions, x02 and x03, are

x02;3 ¼�Q6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðQ2 � 4PR

p2P

; (4)

where 2 and 3 correspond to theþ and – sign, respectively.

Here, P ¼ ða2

12þ d

3Þc; Q ¼ ða2

8þ d

2Þb and R ¼ ð1� dÞ a

2.

These fixed points are shown in Fig. 1(a) with symbols dia-

mond (blue �), circle (black �), and square (red �), respec-

tively, as a function of parameter d. This figure clearly

shows that the fixed point (x01, y0¼ 0, z0¼ 0) always exists,

while (x02,3, y0¼ 0, z0¼ 0) are created near d¼ 0.25 (see

Fig. 1(b) as an expansion of dotted box B of (a)). Also shown

in this figure are the maxima of x (xmaxima) obtained numeri-

cally using fourth-order RK4 algorithms with step size 0.01.

This figure also serves as the bifurcation diagram, where vio-

let and green dots correspond to the branches b1, b2, and b3,

respectively, as obtained from different initial conditions.

The expansion of the dashed Boxes B, C, D, and E are shown

in Figs. 1(b)–1(e), respectively, which is discussed in details

in Sec. IV. We have also shown the time series for a chaotic

case (d¼ 3.415), for a short time span in Fig. 2. It was

observed that the time series were identical for time step size

0.01 and 0.001 as shown in Fig. 2. This justifies using a time

step of size 0.01 in the fourth order RK4 algorithms.

IV. STABILITY ANALYSIS

In order to understand some of the different transitions

in Fig. 1, we consider the linear stability analysis at the fixed

points of Eq. (2), which gives the cubic characteristic equa-

tion as

k3 þ Dk2 þ Ckþ Ad þ B ¼ 0; (5)

where A ¼ � a2þ bx0i þ cx2

0i; B ¼ a2þ a2b

4x0i þ a2c

4x2

0i; C ¼ 1

þ a2

4þ abx0i þ acx2

0i, and D ¼ aþ bx0i þ cx20i. Based on nu-

merical calculation of eigenvalues, k, we have shown sym-

bols in Fig. 1 as filled and unfilled, which correspond to the

stable and unstable fixed points, respectively. This clearly

shows that there are changes of the stability of fixed points

within the dotted boxes B, C, and D. The expansions of these

dotted boxes are shown in Figs. 1(b)–1(d), respectively.

We observe the creation of two fixed points, x02 (unsta-

ble) and x03 (stable), as d is increased beyond 0.25 as indi-

cated by arrow in Fig. 1(b). This indicates the possibility of

saddle-node bifurcation. In order to confirm it analytically,

the model Eq. (2) can be recast into the normal form as

P xþ Q

2P

� �2

� Q2

4P2� R

� �¼ 0:

For further simplification, we can write it as

y2 þ a ¼ 0;

where a ¼ ðR� Q2

4PÞ=P and y ¼ xþ Q2P. This represents the

normal form of saddle node bifurcation13 and clearly indi-

cates that two fixed points are created at a¼ 0 (for a< 0,

there are no fixed points; whereas for a> 0, there are two

fixed points), which correspond to d¼ 0.25 and

x02¼ x03¼�Q/2P¼ 30.5. The analytical and numerical

agreements of values confirm the saddle-node bifurcation.

Note that there co-exists a periodic orbit as well in this range

of parameters as denoted by bifurcation branch b1 (see

Fig. 1(a)).

Shown in Fig. 1(b) is an enlarged view of marked box Bof (a), which shows that fixed points x01 and x02 are crossing

each other at d¼ 1 and interchanging their stability. This

indicates the possibility of transcritical bifurcation.13 The

normal form of this bifurcation can be obtained by consider-

ing Eq. (2)

a

2� ad

2

� �xþ Qx2 þ Oðx3Þ ¼ 0;

where O(x3) shows the higher order terms. After dropping

higher order term, it can be approximated as the standard

form of transcritical bifurcation given by

FIG. 1. (a) The x-component of fixed points, x0i, where i¼ 1 (blue �), 2

(black �), and 3 (red �), and the maxima of x as a function of parameter d.

(b)–(e) are the expansion of dotted boxes B, C, D, and E, respectively, of (a).

022311-3 Wharton et al. Phys. Plasmas 21, 022311 (2014)

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rxþ x2 ¼ 0;

where r ¼ a�ad2Q . This indicates the bifurcation at parameter

d¼ 1 corresponding to r¼ 0 and at point x01¼ x02¼ 0, which

is in agreement with the numerically calculated values.

Hence, this confirms it as a transcritical bifurcation.

An expanded view of box D of Fig. 1(a) is shown in

Fig. 1(d). Here, fixed point x03 changes its stability as shown

by the marked arrow. The eigenvalues calculation (not

shown here) shows that real parts of complex eigenvalues

change sign, i.e., real parts of a pair for complex eigenvalues

move from left to right in complex plane. This indicates that

it is a Hopf bifurcation, where stable fixed point becomes

unstable and a limit cycle (corresponding to branch b2) is

created. Note that this bifurcation occurs in presence of other

co-exiting attractors, periodic branch b3, and stable fixed

point x02.

Fig. 1(e) shows another expanded figure of box E, where

a chaotic branch b3 suddenly disappears to a limit cycle

(branch b2) around d� 4.282 (as indicated by arrow). This

FIG. 2. Chaotic time series x(t), x0(t), and x00(t) using time step size (a) 0.01 and (b) 0.001.

FIG. 3. The left and right panels show

the presence of coexisting attractors

and their corresponding basins at pa-

rameters (a)–(b) d¼ 0.27, (c)–(d)

d¼ 3.2, and (e)–(f) d¼ 4.28.

022311-4 Wharton et al. Phys. Plasmas 21, 022311 (2014)

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transition occurs in presence of another fixed point, (x02)

attractor. However, the exact reason for disappearance is still

to be explored.

As we have seen from Fig. 1 that there are many co-

existing attractors and they change with change in parameter d.

In order to visualize these different co-existing attrac-

tors, shown in Fig. 3, are the coexisting attractors (left panel)

and their corresponding basins (right panel). The basins are

calculated using 50 000 different initial conditions in phase

plane ðxic; x0icÞ for fixed x00ic ¼ 0, where subscript ic stands for

initial conditions. At d¼ 0.27 (after saddle-node bifurcation

in Fig. 1(b)), the coexisting attractors and hence their corre-

sponding basin plots are shown in Figs. 3(a) and 3(b),

respectively. Here, the coexisting attractors are two stable

fixed points (x01, x03) and a limit cycle and their basins are

indicated by blue, black, and green regions, respectively.

The presence of coexisting attractors, at d¼ 3.2, and the cor-

responding basin plots, where one stable fixed point x01 and

two limit cycles (corresponding to branches b2 and b3) exist

are shown in Figs. 3(a) and 3(b), respectively. The presence

of mixed dynamics (at d¼ 4.28): one stable fixed point, one

limit cycle, and one chaotic attractor are shown in Fig. 3(e).

The corresponding basin plots are shown in Fig. 3(f). Note

that the color of attractors and their corresponding basins are

the same. Also note that the basin boundaries for all these

attractors are found to be smooth.

An expanded view of the bifurcation diagram of box Fcorresponding to Fig. 1(a) is shown in Fig. 4(a). It shows

xmaxima in the range of control parameters d � (3, 4.5). The

corresponding plot showing Lyapunov exponents is shown

in Fig. 4(b). These figures suggest that there is chaotic

motion, which appears due to either period doubling bifurca-

tion (near P1, P2, P3, etc.), intermittency (I1, I2, I3, etc) or cri-

sis (C1, C2, C3, etc) and have complicated structures in

parameter space.

The sudden appearance and disappearance of the

branches b1, b2, and b3 (Fig. 1) show the presence of inter-

esting nonlinear phenomena arising out of the jerk equation.

Fig. 5 shows the phase space plots for increasing control

parameter d values. Preliminary experimental studies show a

similar trend on increasing magnetic field30 in the concerned

plasma system. Our present study suggests that at the given

set of parameters, possibility of other coexisting attractors is

also possible. This indicates that initial conditions in experi-

mental systems must be carefully chosen to get appropriate

dynamics. Also note that basins of these different oscillations

are quite well separated and hence small strength of noise

will not affect attractor dynamics.

FIG. 4. (a) An expanded view of bifurcation diagram of box F of Fig. 1 and

corresponding (b) largest two Lyapunov Exponents as a function of parame-

ter d. See text for details.

FIG. 5. (a) Phase space plots for control parameter d value (a) 2.5, (b) 3.2, (c) 3.37, (d) 3.415, (e) 3.5, (f) 3.8, (g) 4.25, and (h) 4.5.

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V. CONCLUSION

To summarize, we have obtained and studied the third

order nonlinear differential equation, which is used to model

chaotic electrostatic ion cyclotron oscillations. One interest-

ing feature of this equation is that it does not require an

external forcing term to obtain chaotic behaviour as needed

in the second order nonlinear ODE studied earlier. The equa-

tion obtained is studied analytically and numerically and the

results obtained are explained. It is noted that the results

obtained here are similar to those seen in preliminary experi-

ments on magnetized plasma, i.e., on changing the magnetic

field, a transition from regular to chaotic oscillations were

observed.30 The coexistence of stable equilibria, limit cycles,

and strange attractors is an interesting result, which implies

the possibility of multistability in the modeled experimental

system and studies are currently being done to explore the

same. A recent work by Sprott et al.36 shows a similar result

in a simpler system. Further experimental investigation is

needed to study how well the jerk equation models chaotic

electrostatic ion cyclotron oscillations in magnetized plasma.

The results obtained here provide an insight into chaotic

electrostatic ion cyclotron oscillations and at the same time,

the nonlinear dynamical results obtained from this equation

seems to be of immense importance.

ACKNOWLEDGMENTS

The authors would like to thank the Director, SINP for

his support. A.P. would like to thank the DST, Government

of India for financial support. A.P. also acknowledges the fi-

nancial support from the DU-DST PURSE grant. The

authors would also like to thank the referee for his valuable

suggestions towards improvements of the paper.

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