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 ODEs Obey superposition principle Frequency is independent of the amplitude at all times Have nonlinear forces acting on them are modeled by nonlinear ODEs Dont 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 : Ndimensional 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 < 0 the orbit attracts to a stable fixed point or stable periodic orbit. = 0 the orbit is a neutral fixed point (or an eventually fixed point). > 0 the orbit is unstable and chaotic. For a 3-dim system we have 3 exponents: Chaos: 1 > 0 2 = 0, where | 3| > | 1|=> 1 + 2 + 3 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 = 1k, C1 = 4.7nF, R2 = 994k, 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 Chuas 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 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 Chuas Diode Chuas Diode A Nonlinear Resistor (N R ) NRNR Chaos in Chuas Oscillator Chaos in Chuas Oscillator Single scroll Chaotic attractor Phase PortraitTime waveform Power spectrum Chaos in Chuas Oscillator Double scroll Chaotic attractor Phase PortraitTime waveform Power spectrum Chaos in Chuas Oscillator Lorenz OscillatorRssler Oscillator HindmarshRose 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