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Bifurcation and chaos Bifurcation and chaos Outline Outline of Topics What is bifurcation? Bifurcation and chaos Some concepts Bifurcation examples in MATLAB Shan He What is chaos School for Computational Science Some concepts about chaos University of Birmingham Chaos examples in MATLAB Module 06-23836: Computational Modelling with MATLAB MATLAB toolboxes Assignment Bifurcation and chaos Bifurcation and chaos What is bifurcation? What is bifurcation? Some concepts Some concepts What is a bifurcation? Classification of bifurcations ◮ Local bifurcations : a parameter change causes the stability of an equilibrium to change. Including: ◮ Bifurcation : a small smooth change of some parameter ◮ Saddle-node (fold) bifurcation values (the bifurcation parameters) of a system causes a ◮ Transcritical bifurcation sudden ‘qualitative’ or topological change in its behaviour ◮ Pitchfork bifurcation ◮ Bifurcations occur in both continuous systems, e.g., ODE and ◮ Hopf bifurcation discrete systems. ◮ Period-doubling (flip) bifurcation ◮ Codimension : the number of parameters which must be ◮ Global bifurcations : occurs when ‘larger’ invariant sets, such varied for the bifurcation to occur. as periodic orbits, collide with equilibria. Cannot be detected purely by a stability analysis of the equilibria. ◮ We will focus on local bifurcations in this lecture. Bifurcation and chaos Bifurcation and chaos What is bifurcation? What is bifurcation? Bifurcation examples in MATLAB Bifurcation examples in MATLAB Example: Transcritical bifurcation Example: Saddle-node bifurcation ◮ Local bifurcations. ◮ Local bifurcations. ◮ There is always at least a fixed point exists for all values of a ◮ parameter and is never destroyed. Also known as tangential bifurcation or fold bifurcation. ◮ ◮ A fixed point interchanges its stability with another fixed Two equilibria collide and annihilate each other. point as the parameter is varied ◮ An two-dimensional Saddle-node bifurcation: ◮ An one-dimensional Transcritical bifurcation: dx 2 dt = α x dy − dx 2 ( = y = αx x dt − dt − Bifurcation and chaos Bifurcation and chaos What is bifurcation? What is bifurcation? Bifurcation examples in MATLAB Bifurcation examples in MATLAB Example: Period-doubling (flip) bifurcation Example: Hopf bifurcation ◮ Local bifurcations. ◮ In discrete system: the system switches to a new behavior ◮ Local bifurcations. with twice the period of the original system. ◮ Also known as Poincar´e-Andronov-Hopf bifurcation. ◮ In continuous system: when a new limit cycle emerges from ◮ A spiral point of a dynamical system switch from stable to an existing limit cycle, and the period of the new limit cycle is unstable, or vice verse, and a periodic solution appears. twice that of the old one. ◮ We will come back to this in our next lecture. ◮ We will see an example of period-doubling bifurcation in a discrete system. Bifurcation and chaos Bifurcation and chaos What is chaos What is chaos Some concepts about chaos Some concepts about chaos Concepts about chaos Relationships between bifurcation and chaos Chaos or chaotic behaviour is a tricky definition. ◮ Chaos by dictionary : a state of disorder. ◮ In chaos theory: “chaotic systems are distinguished by sensitive dependence on initial conditions and by having evolution through phase space that appears to be quite random” – Wolfram Mathworld It will be too abstract if I simply list all the points here. Let me ◮ Several properties: explain this later using the MATLAB example of Logistic Map. ◮ Having a dense collection of points with periodic orbits, ◮ Being sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states), a property sometimes known as the butterfly effect, and ◮ Being topologically transitive: points in the system eventually move under iteration from one arbitrarily small open set to any other ◮ I will illustrate these properties using MATLAB examples. Bifurcation and chaos Bifurcation and chaos What is chaos What is chaos Chaos examples in MATLAB Chaos examples in MATLAB Logistic Map Logistic Map and chaos ◮ Proposed by the biologist Robert May. ◮ Very simple non-linear dynamical difference equations derived from logistic equation. ◮ We only care 3 .57 r 4, where the system begins to ≤ ≤ ◮ With specific parameters, it display chaotic behaviour. exhibit chaotic behaviour. ◮ The equation: ◮ Chaotic behaviour: slight difference in the initial value of x(0) yield dramatically different results over time. xn+1 = rx n(1 xn) ◮ Islands of stability: for r = 1 + √8 3.83, the system shows − ≈ non-chaotic behavior where 0 xn 1 and r > 0 is known as the ”biotic ≤ ≤ potential” or bifurcation parameter. ◮ In biology, xn represents the ratio of existing population to the maximum possible population at year n. Bifurcation and chaos Bifurcation and chaos What is chaos What is chaos Chaos examples in MATLAB Chaos examples in MATLAB Bifurcation diagram Feigenbaum constant ◮ Bifurcation diagram shows the possible long-term values (equilibria or periodic orbits) of a system as a function of a ◮ The first Feigenbaum constant: bifurcation parameter in the system. δ = 4 .66920160910299067185320382 . ◮ Bifurcation diagram of Logistic map: the bifurcation ◮ Calculated from: parameter r is shown on the horizontal axis of the plot and b b − the vertical axis shows the possible long-term population δ n n 1 = lim→∞ − values of the logistic function. n bn+1 bn − n ◮ Shows the forking of the possible periods of stable orbits from where bn is the point at which a period 2 -cycle appears. 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a ◮ The ratio of the intervals between the bifurcation points period-doubling bifurcation. The ratio of the lengths of approaches Feigenbaum’s constant. successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant. Bifurcation and chaos Bifurcation and chaos What is chaos What is chaos Chaos examples in MATLAB Chaos examples in MATLAB Relationships between bifurcation and chaos Lorenz chaotic attractor Lorenz chaotic attractor: ◮ Bifurcation: sensitivity to parameters ◮ Discovered by Edward N. Lorenz, a MIT mathematician. ◮ Chaos: sensitivity to initial conditions ◮ Used to model fluid flow of the earth’s atmosphere. ◮ The transition to chaos is preceded by infinite levels of ◮ The foundation of today’s chaos theory. bifurcation ◮ A 3-dimensional dynamical system that exhibits chaotic flow. ◮ The infinite bifurcations preceding the transition to chaos are ◮ It is deterministic. characterised by the Feigenbaum number. ◮ The system exhibits chaotic behavior and displays a strange attractor. Bifurcation and chaos Bifurcation and chaos What is chaos What is chaos Chaos examples in MATLAB Chaos examples in MATLAB Strange attractors Lorenz equations ◮ Structure has fractal dimension. dx dt = σ(y x) ◮ Do NOT know exactly the location of the attractor: two dy − dt = x(ρ z) y points on the attractor that are near each other at one time − − dx = xy + βz will be arbitrarily far apart at later times. dt ◮ Strange attractors of a systems never close to themselves: the where σ, ρ, β > 0 σ is called the Prandtl number and ρ is called 8 motion of the system never repeats (non-periodic). the Rayleigh number. Usually ρ = 10, σ = 3 and when ρ = 28 the system exhibits chaotic behaviour. Bifurcation and chaos Bifurcation and chaos What is chaos What is chaos Chaos examples in MATLAB Chaos examples in MATLAB Lorenz attractor in MATLAB Properties of the Lorenz attractor 1 function dy=myLorenz(t,x) ◮ Nonlinearity : the two nonlinearities are xy and xz 2 % The RHS of the Lorenz attractor ◮ 3 sigma = 10; Symmetry : Equations are invariant under ( x, y) ( x, y), → − − 4 r = 28; which indicates that if ( x(t), y(t), z(t)) is a solution, so is 5 b = 8/3; ( x(t), y(t), z(t)) − − 6 dy =[ sigma *(x(2)-x(1)); ◮ Volume contraction : the Lorenz system is dissipative, i.e., 7 r*x(1)-x(2)-x(1)*x(3); volume in phase space shrinks exponentially fast at rate of 8 x(1)*x(2)-b*x(3)]; (ρ + 1 + β) 9 end Bifurcation and chaos Bifurcation and chaos What is chaos MATLAB toolboxes Chaos examples in MATLAB Applications of Bifurcation and Chaos theory to biology MATLAB toolboxes ◮ In the context biology, bifurcation theory describes how small changes in an parameter can cause qualitative change in the behavior of the biological system. ◮ The ability to make dramatic change in system output is often essential to organism function, and bifurcations are therefore ◮ MATLAB Adventures in Bifurcations & Chaos (ABC++) ubiquitous in biological networks such as the switches of the ◮ Chaotic Generators Demo cell cycle. ◮ Example: Biochemical switches in the cell cycle. ◮ Chaos in biological systems: ◮ Population growth in ecology. ◮ Neural systems, e.g., action potentials in neurons – Hodgkin-Huxley model. Bifurcation and chaos Assignment Assignment Use MATLAB to find bifurcation points and the calculate Feigenbaum number..
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