Nonlinear Systems Lecture 2 PDF

Nonlinear Systems Johan Suykens KU Leuven, ESAT-STADIUS Kasteelpark Arenberg 10, B-3001 Leuven, Belgium Email: [email protected] http://www.esat.kuleuven.be/stadius/ Lecture 2 (Figures: S. Strogatz, Nonl...

Nonlinear Systems Johan Suykens KU Leuven, ESAT-STADIUS Kasteelpark Arenberg 10, B-3001 Leuven, Belgium Email: [email protected] http://www.esat.kuleuven.be/stadius/ Lecture 2 (Figures: S. Strogatz, Nonlinear Dynamics and Chaos, Westview Press, 2015)  Vector field Consider the vector field: f (x) = [f1(x1, x2); f2(x1, x2)]. One can evaluate this e.g. at a grid of points in the state space and further complement it with simulations from a number of initial conditions to get an idea of the behaviour in the state space. 1  One dimensional flows Example: ẋ = sin x. One has the analytic solution: t = log | cosecx 0 +cotx0 cosecx+cotx |. However, within the scope of this course we are rather interested in qualitative properties, in order to get insight into the behaviour. In this case there are alternating (locally) stable and unstable equilibrium points. 2  Equilibrium points Equilibrium points (fixed points) x∗ are all points satisfying ẋ = 0. Often one proceeds by 1. characterizing all equilibrium points x∗ that satisfy ẋ = 0 (one looks for all points x∗ satisfying f (x∗) = 0) 2. analyzing the stability properties of these equilibrium points (local/global stability, half-stable, unstable). 3  One dimensional flows Solutions in time for different initial conditions x(0): 4  One dimensional flows Example with a locally stable and an unstable equilibrium point: 5  Example of population growth The linear model Ṅ = rN with N denoting the population size and r > 0 the growth factor is not realistic. A more relevant nonlinear model [Verhulst, 1838] is N Ṅ = rN (1 − ) K with K the carrying capacity. Note that only N ≥ 0 values are physically relevant here. 6  Stability analysis by linearization (1) Consider ẋ = f (x) with x ∈ R. Equilibrium points x∗ satisfy f (x∗) = 0. Let us consider a small perturbation η(t) around a particular equilibrium x∗, which can be expressed by η(t) = x(t) − x∗. We want to obtain now an equation describing the evolution of η(t) over time. This is done by taking the time derivative of η(t): d η̇ = (x − x∗) = ẋ dt = f (x) = f (x∗ + η(t)). 7  Stability analysis by linearization (2) where we take the Taylor expansion df (x) 2 f (x∗ + η(t)) = f (x∗) + η + O(η ). dx x ∗ Because f (x∗) = 0 and ignoring the higher order terms one obtains df (x) η̇ = η dx x∗ which is a linear system. In conclusion there are 3 possibilities: ′ f (x∗) > 0: x∗ is unstable ′ f (x∗) < 0: x∗ is locally stable ′ f (x∗) = 0: one cannot make any conclusion in this case! 8  ′ Examples of systems with f (x∗) = 0 9  Local stability analysis for the logistic equation Logistic equation: N Ṅ = f (N ) = rN (1 − ). K Equilibrium points: N ∗ = 0 and N ∗ = K. ′ One has f (N ) = r − 2rN/K. Hence ′ ′ f (0) = r and f (K) = −r with r > 0. Hence N ∗ = 0 unstable and N ∗ = K stable. 10  Existence and uniqueness of solutions (1) Solutions to nonlinear systems are not necessarily unique! Here is an example: ẋ = x1/3 with x(0) = 0. For this system x(t) = 0 is a solution x(t) = ( 23 t)3/2 is a solution. 11  Existence and uniqueness of solutions (2) Existence and uniqueness Theorem: Consider the initial value problem ẋ = f (x), x(0) = x0. Suppose ′ f (x) and f (x) are continuous on an open interval R x0 ∈ R. Then the inital value problem has a solution x(t) on some time interval (−τ, τ ) around t = 0, and the solution is unique. 12  Existence and uniqueness of solutions (3) Consider the example: ẋ = 1 + x2. It has the solution x(t) = tan t. However, the solution only exists for −π/2 < t < π/2 ! Moreover, this system reaches infinity in finite time! (so-called blow-up phenomenon). 13  Impossibility of oscillations In first order system (in continuous time) one cannot have periodic solutions. 14  Potentials (1) Potential function V (x): dV (x) ẋ = f (x) = −. dx Equilibrium points correspond to maxima or minima of V (x). Property: V (t) decreases along trajectories, because dV (x(t)) dV dx =. dt dx dt dV dV =.(− ) dx dx dV 2 = −( ) dx ≤ 0. 15  Potentials (2) Examples: ẋ = −x ẋ = x − x3 16  Optical illusion Figure: Ulijn & St.Amant, May 2000 17  Bifurcations System: ẋ = f (x). Assume that the system is depending on a parameter r. ẋ = f (x; r). We would like to understand now how the properties of the system are changing when we are changing the parameter r. More specifically, we will be interested in understanding how the equilibrium points are changing when we change the parameter r. 18  Saddle-node bifurcation (1) Example: ẋ = r + x2 19 Saddle-node bifurcation (2) 20 Bifurcation diagram 21  Saddle-node bifurcation (3) Example: ẋ = r − x2 leads to a so-called blue sky bifurcation. 22  Normal forms (1) Example systems like ẋ = r −x2 or ẋ = r +x2 are prototypical for all saddle- node bifurcations. Therefore they are considered to be normal forms. Take for example another example which leads to a similar form: ẋ = r − x − e−x = r − x − [1 − x + x2/2! +...] = (r − 1) − x2/2 +... 23  Normal forms (2) Regarding f as a function of both x and r and examining the behaviour of ẋ = f (x, r) near the bifurcation at x = x∗ and r = rc, one obtains ẋ = f (x, r) 2 ∗ = f (x , rc) + (x − x∗) ∂f ∂x + (r − rc) ∂f ∂r + 1 2 (x − x∗)2 ∂∂xf2 ∗ +... (x∗,rc ) (x∗,rc ) (x ,rc ) One has f (x∗ , rc) = 0 because x∗ is a fixed point ∂f ∂x = 0 by the tangency condition of a saddle-node bifurcation. (x∗ ,rc ) Therefore, one obtains ẋ = a(r − rc) + b(x − x∗)2 +... ∂f 2 1∂ f where a = ∂r ,b= 2 ∂x2. (x∗ ,rc ) (x∗ ,rc ) 24  Transcritical bifurcation (1) Example: ẋ = rx − x2 25  Transcritical bifurcation (2) Bifurcation diagram: 26 Laser threshold 27  Pitchfork bifurcation (1) Normal form of supercritical pitchfork bifurcation ẋ = rx − x3 28  Pitchfork bifurcation (2) Bifurcation diagram: 29  Pitchfork bifurcation (3) Potentials V (x): 30  Subcritical pitchfork bifurcation (1) System ẋ = rx + x3 with bifurcation diagram: 31  Subcritical pitchfork bifurcation (2) Taking an additional stabilizing term (−x5) gives ẋ = rx + x3 − x5 giving the bifurcation diagram: 32  Subcritical pitchfork bifurcation (3) This gives the possibility of jumps and hysteresis when r is slowly varied: 33  Imperfect bifurcations and catastrophes Consider the system ẋ = h + rx − x3 with an additional imperfection parameter h. 2r p Saddle-node bifurcations occur at h = ±hc(r) with hc(r) = 3 r/3. The two bifurcation curves meet at (r, h) = (0, 0), called a cusp point. 34 Bifurcation diagrams (1) 35 Bifurcation diagrams (2) 36 Cusp catastrophe surface 37 Biological example: insect outbreak 38  Flows on the circle Consider a previous example but with θ on the circle: θ̇ = sin θ 39  Uniform versus Nonuniform oscillator Uniform oscillator: θ̇ = ω with ω a constant gives the solution θ(t) = ωt + θ0. Nonuniform oscillator: θ̇ = ω − a sin θ 40 Nonuniform oscillator 41  Nonuniform oscillator Examples of nonuniform oscillators: Electronics (phase-locked loops) Biology (oscillating neurons, firefly flashing rhythm) Condensed-matter physics (Josephson junction) Mechanics (overdamped pendulum driven by a constant torque) 42  Nonuniform oscillator: example Let θ denote the phase of a firefly’s flashing rhythm with θ = 0 the instant at which a flash is emitted. In the absence of stimuli, the behaviour satisfies θ̇ = ω. Assume now a periodic stimulus with phase Θ satisfying Θ̇ = Ω, where Θ = 0 corresponds to the flash of the stimulus. A simple model for the overall behaviour is θ̇ = ω + A sin(Θ − θ) with a constant A > 0. The behaviour and synchronization is studied for the phase difference φ = Θ − θ, resulting into a nonuniform oscillator equation φ̇ = Θ̇ − θ̇ = Ω − ω − A sin φ with state variable φ. Depending on the parameters one may obtain synchronization with stable fixed point φ∗ = 0, or also phase locking with φ∗ > 0. 43