# Nonlinear plot for linear complex polynomial

I have a linear complex polynomial. When I plot the polynomial,the real part generates a linear plot but the imaginary part generates me a highly nonlinear plot with bifurcations. My question is

1. Is it possible that a linear polynomial gives a real and a nonlinear plot?

2. How can I remove the nonlinearity so as to generate an approximate linear plot for the complex polynomial?

• What specifically is the polynomial?
– MPW
Commented Aug 13, 2020 at 19:34
• And how are you plotting it? Commented Aug 13, 2020 at 21:12

The imaginary part of the linear equation which gives nonlinear plot is as given below:

Blockquote

y = (-q11*(1/xi - I k) - x*(q22*(1/xi + I k) - q33*(1/xi + I k)(I k - 1/xi) - q44(1/xi^2 + k^2) - q55*(1/xi - I k)))/(p22 + p33*(I k - 1/xi) - p55*(1/xi^2 + k^2))

I'm plotting it using mathematica 9.0 The code is as given below:

Blockquote

omegaJ = 4*10^-15;
lambdaJ = 10^12;
zion = 1.602*10^-19;                           (*ion

  charge*)

e = 1.602*10^-19;                                     >     (*electron \

  charge*)

teqlb = 10^6;                                   (*

  equilibrium temp *)

melec = 9.1094*10^-31;         (* mass of electron *)
ionFreq =
1.00*10^1;             (* rate of electron impact
ionization of \
neutral atoms*)
freqID = 5.00*10^-3;            (* ion-dust
attachment frequency*)
freqED = 1.00*10^2;              (* electron-dust
attachment \
frequency*)
rdust = 37.5*10^-6;              (* dust radius *)
mion = 2*10^-14;                        (*ion mass *)
mnd = 4.0*10^-16;                     (*-ve dust
mass*)

grav = 6.67*10^-11;           (*gravitational
constant*)
visco = 12*10^-6;
(*viscosity*)
cs = 4*10^-3;                        (*sound speed *)
ne0 = 5*10^3;
(*eqlb \
electron density*)
ni0 = 10^3;
(*eqlb \
ion density*)
nnd0 = 1.39*10^1;
(*eqlb negative \
dust density*)
veff = ionFreq/(2*
ni0)  ;               (* difference between volume
recomination \
rate and stepwise ionization rate *)

znd0 = 10^4;
(*negative dust \
charge*)
qnd0 = -znd0*e;
(*equilibrium dust charge*)

etemp = 10^5;
itemp = 10^4;

Subscript[i, e0] =
Abs[-\[Pi] *rdust^2*e*((8*etemp)/(\>
[Pi]*melec))^0.5*ne0*
Exp[(e*qnd0)/(rdust*etemp)]];          (*electron
current*)
Subscript[i, i0] =
Abs[\[Pi]*rdust^2*e*((8*itemp)/(\[Pi]*mion))^0.5*
ni0*(1 - (e*qnd0)/(rdust*itemp))]  ;            (*Ion >     current *)
ndCf = 10^4*
e;                                        (*-ve dust
charging \
frequency*)

c = (-freqED + ionFreq - 2*ne0*veff)/omegaJ;
d = (3*visco*omegaJ)/(melec*ne0*lambdaJ*xi^2) +
(visco*omegaJ*k^2)/(
melec*ne0*lambdaJ);
h = ((3*visco)/(lambdaJ*nnd0*mnd*xi^2))*omegaJ;

gamma = (4*\[Pi]*e^2*lambdaJ^2)/teqlb ;
beta = (qnd0*nnd0)/e;

a1 = teqlb/(mion*cs^2);
a2 = -zion*ni0*omegaJ*a1;
a3 = -zion*ni0*(2*veff*ni0 - ionFreq);
a4 = -(beta*Subscript[i, e0])/qnd0;
a5 = (beta*Subscript[i, i0]*(2*veff*ni0 -
ionFreq))/qnd0;
a6 = (beta*Subscript[i, i0]*omegaJ*a1)/qnd0;
r5 = (qnd0*teqlb*omegaJ)/(e*mnd*cs);
r6 = (qnd0*teqlb)/(e*lambdaJ*mnd);
r8 = -(1/(melec*lambdaJ) +
(teqlb*qnd0*omegaJ)/(e*mnd));
r9 = (2*teqlb*k)/(melec*lambdaJ);
r10 = (2*teqlb*k)/(lambdaJ*mnd);

d1 = h*d*c*freqID*ndCf;
d6 = r10*c*d*freqID*ndCf;
d11 = h*d*c*ndCf;
d15 = c*d*r10*ndCf;
d16 = d*c*freqID;

e4 = d*r10*freqID*ndCf - c*d*r10*ndCf*omegaJ -
c*d*r10*freqID*omegaJ -
c*r10*cs*omegaJ*freqID*ndCf;
e11 = d*r10*ndCf - c*d*r10*omegaJ -
c*r10*ndCf*cs*omegaJ;
e13 = d*freqID - d*c*omegaJ - c*freqID - cs*omegaJ;
p11 = a2*d15 + a6*r10*c*d;
p22 = a2*d11 + a6*h*d*c ;
p33 = r8*h*a3*ndCf - r8*h*ne0*freqID*ndCf +
r8*h*a4*freqID +
r8*h*a5 + beta (r5 + r6)*d16*omegaJ;
p44 = r8*r10*a3*ndCf - r8*r10*ne0*freqID*ndCf +
r8*r10*a4*freqID +
r8*r10*a5;
p55 = d1/gamma;
p66 = d6/gamma;

q11 = -beta*(r5 + r6)*d16*ndCf;
q22 =  a2*e11 + a6*r10*d - a6*c*cs*omegaJ*r10;
q33 = r8*r10*a3*omegaJ + r8*r10*ne0*freqID*omegaJ +
r8*r10*ne0*ndCf*omegaJ - r8*r10*a4*omegaJ;
q44 = e4/gamma;
q55 = e13*ndCf;

y = Range[1, 10];
x = Range[1, 10];
visco = 13*10^-6;
y = (-q11*(1/xi - I k) -
x*(q22*(1/xi + I k) - q33*(1/xi + I k)*(I k - 1/xi) -
q44*(1/xi^2 + k^2) - q55*(1/xi - I k)))/(p22 +
p33*(I k - 1/xi) - p55*(1/xi^2 + k^2));
y = y*10^-14;
Plot[{Im[y] /. {xi -> 100}, Im[y] /. xi -> 500,
Im[y] /. xi -> 1000}, {k, 0, 100}, PlotRange -> All,
PlotStyle -> {Directive[Red, Thick], Directive[Blue, >     Dashed, Thick],
Directive[Black, Thick, Dashed]}, Frame -> True,
FrameStyle -> Thick,
PlotLegends -> {"K for \[Xi] = 100", "K for \[Xi] =
500",
"K for \[Xi] = 1000"},
FrameLabel -> {"Wave number (K)",
"Imaginary frequency (\!$$\*SubscriptBox[\(\ [CapitalOmega]$$, \
$$i$$]\))"}, RotateLabel -> True, AspectRatio -> 0.5]

Plot[{Im[y] /. {xi -> 10}, Im[y] /. xi -> 50, Im[y]
/. xi -> 100}, {k,
0, 100}, PlotRange -> All,
PlotStyle -> {Directive[Red, Thick], Directive[Blue, >     Dashed, Thick],
Directive[Black, Thick, Dashed]}, Frame -> True,
FrameStyle -> Thick,
PlotLegends -> {"K for \[Xi] = 10", "K for \[Xi] =
50",
"K for \[Xi] = 100"},
FrameLabel -> {"Wave number (K)",
"Imaginary frequency (\!$$\*SubscriptBox[\(\ [CapitalOmega]$$, \
$$i$$]\))"}, RotateLabel -> True, AspectRatio ->
0.5]