University of Delhi, Delhi, India

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Presentation transcript:

University of Delhi, Delhi, India Route to chaos Awadhesh Prasad University of Delhi, Delhi, India

-- Models -- Functions -- Objects

Length of lightening? Detection of direction of information flow in coupled complex systems

Length of a tree? Detection of direction of information flow in coupled complex systems

Length of coastline

Area of surface?

Examples Linear: Point, Line, Plane, Cube Nonlinear: Sphere, length of a tree Detection of direction of information flow in coupled complex systems

Linear vs Nonlinear Detection of direction of information flow in coupled complex systems

Differential equation: Linear Nonlinear

Differential equation: Linear Nonlinear

Pendulum Individual/single

Pendulum: Linear vs Nonlinear T0=time at vertical which th=0 sn Jacobian Elliptic function

Pendulum: Linear vs Nonlinear T0=time at vertical which th=0

Center Linear Pendulum Damped Undamped Stable T0=time at vertical which th=0 Center Stable

Nonlinear pendulum Undapmed Damped Center/saddle Stable & Unstable T0=time at vertical which th=0 Center/saddle Stable & Unstable

Solutions = ? + Dissipation Nonlinearity

Fixed/stationary points 1D Systems Fixed/stationary points

Fixed/stationary points 1D Systems Fixed/stationary points U S U S U S U S

2D Systems

>2D Systems

Solutions Invariants Attractors

Solutions -- Fixed point -- Periodic -- Quasiperiodic -- Chaotic

Solutions -- Fixed point Nonlinear Linear -- Periodic -- Quasiperiodic -- Chaotic

Fixed point Solutions l=a ib

Periodic Solutions Unstable fixed points

Periodic solutions

Quasiperiodic

Quasiperiodic

Stable/unstable periodic orbits Chaos!!

Chaotic Solutions

Chaotic Solutions

Properties of chaos -- geometrically strange -- temporally irregular -- sensitive to initial conditions Due to UPO?

Unstable Periodic Orbits (UPOs)

Bifurcation Qualitative change in dynamics as parameter varies Sand piling Heartbeat

Chaos to Periodic: Heart Attack Christini D J et al. PNAS 98, 5827(2001)

Montage

Onset of a temporal lobe epileptic seizure Ref. L. Iasemidis

EEG: Epileptic Patient (temporal lobe epilepsy) Preictal ictal postictal minutes

Aim Periodic Chaotic Routes? Chaotic Chaos

Period doubling: Laser

End