Theory of nonlinear dynamic systems Practice 2 Juhász János juhasz.janos@.itk.ppke.hu Szélig Ádám szelig.adam.gyorgy@itk.ppke.hu Goda Márton goda.marton.aron@itk.ppke.hu 2018.06.05.

I. Equation Examine the following equation: Simple harmonic oscillator, LC circuit, "perpetual motion" Constant energy Only the amplitude is changing at setting of Xinit-Yinit’s parameter, no any effect to the characteristic of the curve.

II. Equation Examine the following equation: Damped Harmonic Oscillator, RLC circuit. Changing energy – petering out, relaxation. Xinit-Yinit has effect to the attenuation b: most important parameter, attenuation rate. b < 2, b = 2 , b > 2.

III. Equation Examine the following equation: Excitation of the Classical Harmonic Oscillation, RC circuit, excitation by alternate current (AC). Resonant case Energy increses, the system „explodes” Settings of Xinit-Yinit’s parameters cos(t): excitation factor

IV. Equation Examine the following equation: Same to the previous one, but it has a plus frequency component (omega) in the excitation. The energy is changing sinusoidal because of the Inhomogeneous DE wb : oscillation of angular frequency wk = 1 In the case of III. equation : wb = wk = 1

V. Equation Examine the following equation: Equation of harmonic oscillator. General solution formula: Energy is oscillating

VI. Equation Examine the following equation: Damped system under Harmonic Force, stable periodic orbit in case of infinite time Initial conditions have no effect the stable orbit (in most cases…) Steady Sustained Energy b: greater b parameter results in reaching periodic orbit earlier (system lost the initial energy and only the excitation sustains it)

VII. Equation Examine the following equation: Simple Pendulum Equations: swinging ball, rocking ship The law of conservation of energy

VIII. Equation Examine the following equation: Damped pendulum Energy is petering out, the system relaxes b: defines the speed of relaxation

IX. Equation Examine the following equation: Oscillation with excitation and damping Chaos  Energy ? (remark: pictures were made at Xinit = 1 and Xinit = 1.1 initial conditions – it is well visible the relevant difference at small change)

X. Equation Examine the following equation: Excitation with resonant frequency b=0.05 ω=0.9 Double period

Kapitza's Pendulum I. https://www.youtube.com/watch?v=cjGqxF79ITI

Kapitza's Pendulum II. Inverted pendulum Equation of pendulum: Modified excitation function The system can stay in its the instable fix point

Van der Pol oscillator I. An electronic oscillator is an electronic circuit that produces a periodic, oscillating electronic signal, often a sine wave or a square wave Solution of DE: constant amplitude of sinusoidal signal, where U0 is the amplitude , ω0 is the frequency and ς0 is the phase. Task: Circuit Implementation

Van der Pol oscillator II. Problems: - It depends on the initial conditions (after turning of circuit the amplitude might be change) - In reality it is necessarily a perfect structure to sinusoidal solution and constant amplitude - Constant amplitude oscillator could not be constructed with linear elements only Conclusion: We must ensure constant and stable frequency /amplitude vibration. Furthermore, the system have to reach the constant frequency /amplitude in case of any initial conditions.

Van der Pol oscillator III. Solution:

Tank you for your attention!

Matlab® supplement [X,Y] = meshgrid(x,y) replicates the grid vectors x and y to produce a full grid. equation= @(t,y) [y(2); y(1)]; [1. equ of the system; 2. equ of the system] [t,y]=ode45(equation, [t0,tmax][Xinit,Yinit]); there are other solvers as well, first we try this figure creates figure graphics objects. Figure objects are the individual windows on the screen in which the MATLAB software displays graphical output. plot(x,y,how…) drawing, has many options subplot(m,n,p) (divide the figure m*n parts, draws in the p th region) contour(X,Y,Z), contour(X,Y,Z,n), and contour(X,Y,Z,v) draw contour plots of Z using X and Y to determine the x- and y-axis limits.