Nonlinear Systems

Nonlinear Systems by Thomas McClure I Introduction This paper is written on Nonlinear Systems. II

Nonlinear Systems

Chaotic Dynamics of Nonlinear Systems By S. Neil Rasband, paper, c. 1990 Chapter 1 In this form the system consists of three, first-order differential equations.... The differential equation for a simple pendulum. (p. 3) xdot = f(x,t) (1.1) (p. 2) dx/dtheta = y, dt/dtheta = 1 dy/dtheta = -omega0^2sinx + f(x,t) (p. 3) (p. 17) ... sensitive dependence on initial conditions ... making chaos possible. (p. 18) The Lyapunov Exponent (p. 19) ... the tent map delta_p(x). For mu = 1 we have ... lambda(x) = ln 2 . deta x = &2^n = &e^nlambda ...the logistic map F_mu(x) = mux(1-x) Suppose x(1-x) == -1 x {0.6,2.6} Chapter 2 lambda(x_0) = lim{n,%}{(1/n){sum{i=0,(n-1)}{ln|f'(x_i)|

(2.2) (p.19)

... the Lyapunov exponent lambda was a good indicator of chaos .... (p. 21) To summarize: (1.2.1)

xdotdot + omega0^2sin x = f(x,t)

(1.2.2)

xdot = f(x,t) (1.1) (p. 2) dx/dtheta = y, dt/dtheta = 1 dy/dtheta = -omega0^2sinx + f(x,t) (p. 3)

(1.2.3)

dx/dt = y dy/dt = -omega0^2 sin x + f(x,t)

Chapter 3 Universality Theory

(p. 33)

Count the number of chaos-like words: six Chaos is equivalent with period doubling. sigma(1/2) = 1/alpha

(p. 50)

xdotdot = f_mu(x,t)

(p. 53)

xdotdot = y

(p. 55)

ydotdot = -muy - x^3 + b sin(2 pi t)(p. 55) Experimental Results

(p. 59)

Water (1980)

2 period doublings

Laser (1982)

2 period doublings

dx/dn = mu + x^2 f_mu(x) = x/(1-x)

(3.83) (3.94)

(p. 62)

(p. 66)

Chapter 4 Lyapunov Characteristic Exponent (LCE) lambda_i(x_0) = lim{n,%}{(1/n)ln[mu_i(n)], i = 1,2,..., m

(4.11)

See (2.2) d_L = (1-(lambda_1/lambda_2)

(4.12)

(p. 74)

Chapter 5 xdotdot + %^2 sin x = epsilon(-alpha xdot + f cos omega t) (5.1) xdot = y ydot = -alpha^2 sin x - alphay

(5.8)

(p.87)

xdot = xy + x^2 ydot = -y - x^2y

(5.16)

(p. 90)

(5.17)

(p. 91)

(5.25)

(p. 96)

(5.43)

(p. 99)

(5.46)

(p. 100)

xdot = x^2 ydot = -y xdot = sigma(y - x) ydot = px - y - xz zdot = -beta z + xy xdot = - x^2 ydot = -y + x^2 xdot = -y + xa ydot = x + yz zdot = -z - (x^2 + y^2) + z^2 xdot = -y + x(mu - (x^2 + y^2)) ydot = x + y(mu - (x^2 + y^2)) Chapter 6 xdot_1 + vxdot_2 = -mux_1 + y_1x_2 xdot_2 + vxdot_1 = -mux_2 + y_2x_1 ydot_1 = 1 - x_1x_2

(5.74)

(p. 109)

ydot_2 = 1 - x_1x_2

(6.13)

(p. 114)

(p.133)

xdotdot + %^2 sin x = epsilon(-alpha xdot + f cos omega t) (6.28) xdot = y ydot = -%^2 sin x - alphay + f cos omega t

(6.29)

(p. 133)

xdotdot - (1/2)x(1 - x^2) = -alphaxdot + f cos omega t (6.30)(p.127) Chapter 7 x_(n+1) = ax_n(1 - x_n - y_n) y_(n+1) = bx_ny_n

(7.37)

(p. 143)

(7.40)

(p.148)

(7.43)

(p.153)

x_(n+1) = 1 - ax_n^2 + y_n y_(n+1) = bx_n x_(n+1) = 1 - ax_n^2 + y_n y_(n+1) = -x_n Chapter 8 J_(n+1) = J_n theta_(n+1) = theta_n + 2pi alpha(J_(n+1)

(8.52)

(p. 176)

Chapter 9 lambda(x,omega0) = lim{n,%}{(1/n)sum{i=1,n}(ln alpha_i)/delta t (9.37) (p. 191) (LCE) Chapter 10 https://www.academia.edu/19660215/perfect_secrecy III

Conclusion

This paper is written on Nonlinear Systems. It is obvious that mu is the variable that distinguishes solutions. Also ydotdot and and zdot and ydot are functions of y.

Appendix Observation: (1.2.3)

dx/dt = y dy/dt = -omega0^2 sin x + f(x,t)

Comparing these two systems, it is obvious that the chaotic term is ay. Some of the Rössler attractor's elegance is due to two of its equations being linear; setting z = 0, allows examination of the behavior on the x, y plane. https://en.wikipedia.org/wiki/R%C3%B6ssler_attractor {dx}/{dt} = -y {dy}/{dt} = x + ay as compared to: xdotdot = f_mu(x,t)

(p. 53)

xdotdot = y

(p. 55)

ydotdot = -muy - x^3 + b sin(2 pi t)(p. 55) that is, ydotdot = -x^3 + b sin(2 pi t) - muy = f(x,t) -muy where the chaotic term is -muy.