When dealing with real life systems, we try to interpret the systems qualitatively rather than quantitatively as most of them is nonlinear in behaviour and have extremely complex dynamics. Most fundamental approach is interpreting the differential equation vector field, and then drawing vector fields analogous to flows of the fluid in line. By this phase portrait analysis we can easily say how the system evolves with time. Whereas by exploiting bifurcation diagram we can visualise the transitions due changes in parameters of dynamical system. Metamaterial is man-made material that can be made of nonlinear materials and hence had a nonlinear response to the electromagnetic wave. In addition, exotic properties such as a negative refractive index, metamaterials create opportunities to tailor the phase matching conditions that must be satisfied in any nonlinear optical structure. Here we don’t want to look at the metamaterials as a material scientist rather our main concern is the dynamics of the metamaterial and prediction of different phases that it is passing through.

1. Project Advisor:
Manish Dev Shrimali
PREMASHIS KUMAR
M.SC PHYSICS
2016MSPH005

2. Contents
History of Dynamics
One-Dimensional Flows :
Flows On The Line
Bifurcations
Future Project : Dynamics of
Metamaterial
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3. Dynamics-A Capsule History
In the mid-1600s,Newton invented differential equations.
The breakthrough came with geometric approach of Poincare in the
late 1800s.
Lorenz's discovery of chaotic motion on a strange attractor in
1963 .
Mandelbrot codified and popularized fractals.
Feigenbaum discovered completely different systems can
go chaotic in the same way.
Winfree applied geometric methods of dynamics to biological
oscillations. 3

4.  One dimensional system:
𝒅𝒙
𝒅𝒕
=f(x)
Graphical analysis: Interpreting a differential equation as a vector field.
i. Plot 𝒙 vs 𝒙.
ii. Draw arrows on x axis for corresponding velocity vector.
 Flow on the line: 𝒙 <0 To the left 𝒙 >0 To the right.
Some basic terms: Phase Point , Trajectory, Phase Portrait.
At points 𝒙∗ where
𝒅𝒚
𝒅𝒙
= 0 no flow fixed points STABLE : flow is toward them.
UNSTABLE : flow is away from them.
A reverse construction : Draw the trajectories and extract solutions.
Differential
Equations
Iterated
Maps
Dynamical Systems
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5.  General framework for ODE is provided by the system:
𝒙 𝟏 =𝒇 𝟏(𝒙 𝟏 , ……. ,𝒙 𝒏)
: : : : : : :
𝒙 𝒏 =𝒇 𝒏(𝒙 𝟏 , ……. ,𝒙 𝒏)
 SYSTEM: Damped Harmonic Oscillator m 𝒙+b 𝒙+k𝒙=0.
Trick: Introduce new variables 𝒙 𝟏= 𝒙 and 𝒙 𝟐= 𝒙 i . 𝒙 𝟏 = 𝒙 𝟐
ii . 𝒙 𝟐= 𝒙=−
𝒃
𝒎
𝒙 𝟏 −
𝒌
𝒎
𝒙 𝟐
 So we convert a second order differential into two first order differential equations.
 Nonautonomous systems: Include explicit Time dependence Forced Harmonic Oscillator.
 Again Easy Trick: 𝒙 𝟏= 𝒙, 𝒙 𝟐= 𝒙 𝐚𝐧𝐝 𝒙 𝟑=t. i . 𝒙 𝟏= 𝒙 𝟐
ii . 𝒙 𝟐=−
𝒃
𝒎
𝒙 𝟏 −
𝒌
𝒎
𝒙 𝟐 +
𝑭
𝒎
cos𝒙 𝟑
iii . 𝒙 𝟑= 1.
 An nth order time-dependent equation is a special case of an (n+1) dimensional system.
 Population Growth:
The simplest model for the growth of a population:
𝒅𝑵
𝒅𝒕
= r×N.
 This model predicts exponential growth.
 For populations a certain carrying capacity K,
the growth rate actually becomes negative Logistic Equation:
𝑵=rN(1-
𝑵
𝑲
) 5

6. • Interesting thing about one-dimensional system Dependence on parameters.
As the parameters are varied , fixed points are created or destroyed or
their stability can change .These qualitative changes in dynamics are
called bifurcation.
 Bifurcation point or value= the parameter values at which
change occurs.
Saddle-Node Bifurcation
The most fundamental bifurcation of all.
• Fixed points known as ‘saddles’ and ‘nodes’ can collide and annihilate.
BIFURCATIONS
Parameter is
varied
Two fixed points
move toward
each other
Collide and
coalesce into half-
stable fixed point
Half-stable fixed
point vanishes
soon
 TERMINOLOGY:Conflicting terminology .Also called a fold bifurcation or a Turning point bifurcation or
blue sky bifurcation. Saddle can only exists in two or higher dimension.
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7. 𝒚 𝒚 𝟐 + 𝒏
 The prototypical example of Saddle-node bifurcation:
= n is a parameter.
 Bifurcation occurred at r=0,vector field for r<0 and r>0 are qualitatively different.
TRANSCRITICAL BIFURCATION
 In certain situations , fixed point must exist for all values of parameter . Example : Growth of a
single species.
 Such fixed point may change its stability as parameter is varied..
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8. • The normal form for a transcritical bifurcation:
𝒙=𝒙𝒓 − 𝒙 𝟐
𝒙=𝒙(𝒓 − 𝒙)
𝒙∗
=0 Stable (r<0)
Unstable(r>0)
𝒙∗=r Stable (r>0)
Unstable(r<0)
 Stable and unstable point collide at r=0 and give half stable point but…………..
 Flavour change occurs Fixed point switch their stability after bifurcation. 8

9. Pitchfork Bifurcation
 Commonly found in physical problems that have a symmetry.
 If load increased , the beam may buckle to either the left or the right.
Pitchfork Bifurcation
Supercritical Pitchfork Subcritical Pitchfork
Supercritical Pitchfork
 The normal form of the supercritical pitchfork bifurcation:
𝒙=𝒓𝒙 − 𝒙 𝟑.
 This is invariant under the change x -x.
 What about the term "pitchfork“?
pitchfork trifurcation!!! 9

10.  The normal form of the subcritical pitchfork bifurcation:
𝒙=𝒓𝒙 + 𝒙 𝟑
 Now nonzero fixed points are unstable and exists only below the bifurcation ‘sub critical’
Parameter (r) Saddle-node Transcritical Pitchfork
r > 0 Zero fixed point Two fixed point Three fixed point
r = 0 One fixed point One fixed point One fixed point
r < 0 Two fixed point Two fixed point One fixed point10

11. Metamaterial
A material engineered to have a property that is not found in nature.
wavelength
of the
phenomena
Assemblies
of multiple
elements
 Derive their properties not from the properties of the base materials , Arrangement gives them
smart properties.
 Natural materials only affect E , metamaterials can affect B too.
 First described theoretically by Victor Veselago in 1967.
 Negative-index metamaterials exhibit a negative index of refraction for particular wavelengths.
 John Pendry was the first to identify a practical way.
to make a left-handed metamaterial.
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12.  Negative refractive index Tailor the phase matching conditions
Applications: Absorber,Cloaking devices , Seismic protection , Sound
filtering.
What do metamaterials allow us to do that we couldn’t before?
 Extreme miniaturization of existing optical devices.
Can be customized to support novel properties that currently are not accessible.
What excites me about metamaterials?
 we get to the big questions of applications for these materials and devices. It’s
just wide open.
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13. REFERENCE
[1] Krishnamoorthy,H. N., Jacob, Z., Narimanov, E., Kretzschmar, I., &
Menon, V. M. (2012). Topological transitions in metamaterials. Science,
336(6078), 205-209.
[2] Strogatz, S. H. (2014). Nonlinear dynamics and chaos: with applications
to physics, biology, chemistry, and engineering. Hachette UK.
[3]https://www.nature.com/subjects/metamaterials
[4]https://engineering.stanford.edu/magazine/article/what-are-
metamaterials-and-why-do-we-need-them
[5]Walsh, C. (2017). Industrial Interest in Materials Science.
[6]https://en.wikipedia.org/wiki/Nonlinear_system
[7]https:/en.wikipedia.org/wiki/Metamaterial
[8]https://phys.org/news/2017-08-invisibility-cloak-closer-revealing.html
[9] Thompson, J. M. T., & Stewart, H. B. (2002). Nonlinear dynamics and
chaos. John Wiley & Sons.
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