One dimensional flow,Bifurcation and Metamaterial in nonlinear dynamics
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The document covers the history and principles of dynamical systems and bifurcations, detailing the evolution from Newton's differential equations to modern applications in one-dimensional flows. It introduces key concepts like fixed points and bifurcation types, such as saddle-node, transcritical, and pitchfork bifurcations, alongside discussing the innovative field of metamaterials which are engineered to have unique properties not found in nature. The potential applications of metamaterials, including advanced optical devices and cloaking technologies, highlight their significance in future research.
![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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One dimensional flow,Bifurcation and Metamaterial in nonlinear dynamics
- 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 2
- 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 4
- 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. 6
- 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.. 7
- 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. 11
- 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. 12
- 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. 13
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