Chaos in Electronic Circuits K. THAMILMARAN Centre for Nonlinear Dynamics School of Physics, Bharathidasan University Tiruchirapalli

 To introduce chaos as a dynamical behavior admitted by completely deterministic nonlinear systems.  To give an idea of the equilibrium conditions like fixed points and limit cycles and the routes or transition to chaos admitted by nonlinear systems.  To model nonlinear systems using simple electronic circuits and demonstrate their dynamics in real time for a wide range of control parameters, with the limited facilities available in an undergraduate electronics lab. Aim

 Classification of Dynamical systems  What is Chaos? How does it arise?  What is the characteristic (or) signature of chaos?  Transition to chaos – Routes  How to study or explore chaos?  Study of chaos using nonlinear circuits  Demo – circuit simulators  Conclusion Plan of Talk

Linear Dynamical SystemsNon - Linear Dynamical Systems Dynamical Systems  Have linear forces acting on them  are modeled by linear ODE’s  Obey superposition principle  Frequency is independent of the amplitude at all times  Have nonlinear forces acting on them  are modeled by nonlinear ODE’s  Don’t obey superposition principle  Frequency is not independent of the amplitude. Dynamical systems

Dynamical Systems… Linear Harmonic Oscillator – a linear dynamical system Corresponding two first order equations

Duffing Oscillator – a nonlinear dynamical system Dynamical systems… Corresponding two first order equations

Dynamical systems whether linear or nonlinear are classified as Dissipative systems: whose time evolution leads to contraction of volume/area in phase space eventually resulting a bounded chaotic attractor Conservative or Hamiltonian systems: here chaotic orbits tend to visit all parts of a subspace of the phase space uniformly, thereby conserving volume in phase-space Dynamical systems…

Phase space : N–dimensional geometrical space spanned by the dynamical variables of the system. Used to study time evolution behavior of dynamical systems.

Dynamical systems… Various fixed points

Dynamical systems… Fixed points limit cycles and strange attractors - equilibrium states of dynamical systems Fixed points: are points in phase space to which trajectories converge or diverge as time progresses limit cycles : are bounded periodic motion of forced damped or undamped two dimensional systems strange attractors : characteristic behaviour of systems when nonlinearity is present

Dynamical systems… Limit cycle attractor

Dynamical systems… Quasi-periodic Motion

Dynamical systems… strange attractors

 Chaos is the phenomenon of appearance of apparently random type motion exhibited by deterministic nonlinear dynamical systems whose time history shows a high sensitive dependence on initial conditions  Chaos is “deterministic – randomness”  deterministic - because it arises from intrinsic causes and not from external factors  randomness - because of its unpredictable behavior What is Chaos?

 Chaos is ubiquitous and arises as a result of nonlinearity present in the dynamical systems  It is observed in atmosphere, in turbulent sea, in rising columns of cigarette smoke, variations of wild life population, oscillations of heart and brain, fluctuations of stock market, etc.  Most of the natural systems are nonlinear and therefore study of chaos helps in understanding natural systems How does Chaos arise ?…

Characteristics of Chaos… Chaos is characterized by  extreme sensitivity to infinitesimal changes in initial conditions  a band distribution of FFT Spectrum (power spectrum)  positive values for Lyapunov exponents  fractal dimension

Insensitivity to initial conditions – observed when the absents of nonlinearity Characteristics of Chaos… very high sensitive dependence on initial conditions – observed when nonlinearity is present

Characteristics of Chaos… Power spectrum of a chaotic attractor Power spectrum for a simple sinusoidal wave

Lyapunov Exponent  λ < 0 the orbit attracts to a stable fixed point or stable periodic orbit.  λ = 0 the orbit is a neutral fixed point (or an eventually fixed point).  λ > 0 the orbit is unstable and chaotic. For a 3-dim system we have 3 exponents: Chaos: λ1 > 0 λ2 = 0, where | λ3| > | λ1|=> λ1 + λ2 + λ3 <0 λ3 < 0 Periodic Torus : λ1 = 0 λ2 = 0 λ3 < 0 Periodic Cycle : λ1 = 0 λ2 < 0 λ3 < 0 Largest Lyapunov Exponent:

Routes to Chaos  Period doubling route  Quasi-periodic route  Intermittency route Bifurcations are sudden qualitative changes in the system behavior as the control parameters are varied

Routes to Chaos… Period doubling Bifurcations

Routes to Chaos… Period doubling route to chaos

Routes to Chaos… Experimentally observed Period doubling route

Routes to Chaos…

Quasiperiodic route to chaos

Routes to Chaos… Intermittent bursts and transitions to chaos

Routes to Chaos… Experimental result for Intermittent bursts and transitions to chaos

Rare routes to chaos Period -3 doubling cascade

mathematical modeling and solving the resultant equations of motion by  Analytical methods  Numerical computations Experimental study by  Hardware experiments  Circuit simulations (PSpice, Multisim) How to study chaos? Chaos in dynamical systems can be studied by

 Analytical techniques like Fourier transforms, Laplace transforms, Green’s Function method, etc., are available for linear ODE’s only  Entirely new analytical tools for solving nonlinear ODE’s have to be developed for each system separately  Are often cumbersome and require rigorous knowledge of mathematics How to study chaos?...

Numerical computations  require simple algorithms  are easy to implement using any high level language However  they require enormous computing power and  enormous time to unravel the full dynamics Therefore real time hardware experiments are devised in the laboratory How to study chaos?...

 A circuit may be considered as a dynamical system if it contains energy storing elements such as capacitors and/or capacitors and inductors and linear or nonlinear elements  Using these real time observation and measurements are easily made for a wide range of parameters  Various behaviors like bifurcation and chaos can be observed using oscilloscopes, spectrum analyzers, etc. Study of Chaos Using Electronic Circuits

 Curve tracers and Poincaré section circuits help in easy analysis  Circuit simulators like PSpice provide an "experimental comfort" and help in exploring and confirming strange an unexpected behaviors  Advances in IC technology help in building cheap models that truly reflect the properties of any physical systems Examples are Chua’s circuit, canonical Chua’s circuit and MLC circuit Study of Chaos Using Electronic Circuits…

C HAOS IN N ONLINEAR E LECTRONIC C IRCUITS Minimum requirements for realizing chaos in circuits  One nonlinear element  One locally active resistor  Three energy storage elements for autonomous systems  Two energy storage elements + forcing for nonautonomous systems

Linear and Nonlinear Resistor (a) A two-terminal linear resistor, (b) its (v −i) characteristic curve, (c) a nonlinear resistor and (d) its (v − i) characteristic curve v(t) = R i (t); f R (v,i)=0 Study of chaos using nonlinear circuits…

Linear and Nonlinear Capacitor (a) A two-terminal linear capacitor, (b) its (v − q) characteristic curve, (c) a nonlinear capacitor and (d) its (v − q) characteristic curve Study of chaos using nonlinear circuits…

Linear and Nonlinear Inductor (a) A two-terminal linear inductor, (b) its (φ−i) characteristic curve, (c) a nonlinear inductor and (d) its (φ − i) characteristic curve Study of chaos using nonlinear circuits…

The Fourth Element

LC circuit Study of chaos using nonlinear circuits…

Numerical time waveform of v (t) and phase portrait in the (v − i L ) plane for LC circuit Study of Chaos Using Electronic Circuits…

Forced LCR circuit Study of Chaos Using Electronic Circuits…

Numerical time waveform of v(t) and phase portrait in the (v − i L ) plane of the series LCR circuit Study of Chaos Using Electronic Circuits… Unforced damped oscillation ( F =0, R/L > 0)

Numerical time waveform of v(t) and phase portrait in the (v − i L ) plane. Study of Chaos Using Electronic Circuits… Forced damped oscillation ( F >0, R/L > 0)

Experimental time waveform of v(t) and phase portrait in the (v − i L ) plane. Study of Chaos Using Electronic Circuits… Forced damped oscillation ( F > 0, R/L > 0)

Study of Chaos Using Electronic Circuits… Chaotic Colpitts Oscillator

Study of chaos using nonlinear circuits…

Study of Chaos Using Electronic Circuits…

Oscillator VS with an active RC load composite. V S = 10V/3kHz, R1 = 1kΩ, C1 = 4.7nF, R2 = 994kΩ, C2 = 1.1nF, Q1 = 2N2222A.

Study of Chaos Using Electronic Circuits… limit cyclechaotic attractor V S : 10V/3kHz. y: V (2) = V (C2), 0.5V/div, x: V (5) = V (V S), 2.0V/div

Study of Chaos Using Electronic Circuits… Chaotic Wien's Bridge oscillator

Study of chaos using nonlinear circuits… Clockwise from top left. Fixed point, period 2-T, 4-T, 8-T chaotic oscillations.

Study of chaos using nonlinear circuits… Wein bridge oscillator Wein bridge Chua’s oscillator

Fixed point Study of chaos using nonlinear circuits… 1 bc attractor 4T 1T2T 2 bc attractor Experimental results of Wein bridge oscillator

Study of chaos using nonlinear circuits… Duffing Equation  is an ubiquitous nonlinear differential equation, which makes its presence felt in many physical, engineering and even biological problems.  Introduced by the Dutch physicist Duffing in 1918 to describe the hardening spring effect observed in many mechanical problems

Study of chaos using nonlinear circuits… Duffing equation can be also thought of as the equation of motion for a particle of unit mass in the potential well subjected to a viscous drag force of strength α and driven by an external periodic signal of period and strength f This found to exhibit interesting dynamics like period doubling route to chaos

Study of chaos using nonlinear circuits… Single - well potentialDouble -well potential Single-hump potentialDouble -hump potential

Study of chaos using nonlinear circuits… Analog simulation of Duffing oscillator

Study of chaos using nonlinear circuits… Analog simulation of Duffing oscillator equation – regular dynamics 1 T2 T 3 T 4 T

Study of chaos using nonlinear circuits… Analog simulation of Duffing oscillator – Chaotic dynamics Single band chaosDouble band chaos

v-i characteristic of different Nonlinearities

Circuit realization of Chua’s Diode Chua’s Diode – A Nonlinear Resistor (N R ) NRNR

Chaos in Chua’s Oscillator

Chaos in Chua’s Oscillator…

Single scroll Chaotic attractor Phase PortraitTime waveform Power spectrum Chaos in Chua’s Oscillator…

Double scroll Chaotic attractor Phase PortraitTime waveform Power spectrum Chaos in Chua’s Oscillator…

Lorenz OscillatorRössler Oscillator Hindmarsh–Rose modelLogistic map

Conclusion We have seen  How dynamical systems are classified and their behaviors  What is chaos?, how it is generated?, its characteristics, etc  How systems undergo transitions to chaos  Exploration of chaos in some simple nonlinear circuit like transistor circuits, Colpitts, Wien Bridge, Duffing, Chua and MLC oscillators  Bifurcation and Chaos are observed by Multisim Simulation