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# An Introduction to Nonlinear Systems Chaos APPM 3010

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This 8 page Study Guide was uploaded by Dr. Filomena Hegmann on Thursday October 29, 2015. The Study Guide belongs to APPM 3010 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 82 views. For similar materials see /class/231876/appm-3010-university-of-colorado-at-boulder in Applied Math at University of Colorado at Boulder.

## Reviews for An Introduction to Nonlinear Systems Chaos

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Date Created: 10/29/15

APPM 3010 Review for Exam 2 1 One dimensional maps Be able to nd xed points of simple maps explicitly and deter mine their stability using linear information Be able to identify xed points of more complicated maps graph ically and determine their stability Know the equation satis ed by a period m orbit Given the graph of the mth iteration map7 be able to nd period m solutions Know the stability criterion for period m orbits and be able to apply it Be able to describe in words the period doubling route to chaos Be able to describe how the Lyapunov exponent can indicate a systems tendency towards chaos You need not memorize the de nition of the Lyapunov exponent it Will be given if necessary Given the graph of a map7s Lyapunov exponent versus a parame ter7 be able to indicate where the onset of chaos is expected and where intermittancy is expected 2 Planar linear systems and linearization of planar nonlinear systems 0 Be able to reproduce Figure 528 on page 137 of Strogatz Under stand how the eigenvalues and eigenvectors of a matrix determine the local phase portrait of the corresponding dynamical system Know the di erence between Lyapunov stability and asymptotic stability Be able to provide an example of an equilibrium point that is Lyapunov stable but not asymptotically stable7 and vice versa Know what attracting means and be able to provide examples Be able to provide an example of a xed point that is attracting but not Lyapunov stable Know that Lyapunov stable and attracting together imply asymptotically stable 0 Know when the information from the linearization will predict the correct stability type of an equilibrium point under small nonlinear perturbations 0 Know when the information from the linearization will predict the correct structure type of an equilibrium point under small nonlinear perturbations 0 Be able use information from the linearization to draw a rough picture of the local phase portrait for the corresponding nonlinear system 3 Periodic orbits in planar systems 0 Know the two types of periodic solutions in a planar nonlinear system Be able to give an example of each 0 Be able to state the Poincare Bendixson theorem and know how to apply it Know the implications of this theorem in terms of chaotic behavior 0 Given a nearly decoupled system in polar coordinates7 be able to design a trapping region for the corresponding ow 0 For more complicated ows7 be able to show that a given region is a trapping region 0 Know what nullclines are and how they are useful in determining the global ow 0 Know the hypotheses and conclusion of the Hopf bifurcation the orem 0 Know how to set up a Poincare section for a ow Know how to use the Poincare map to determine if periodic orbits exist and when they do7 their stability type APPM 3010 Review for Final Exam 1 Fundamentals 0 Given an ODE7 possibly a higher order scalar equation andor a nonautonomous equation7 be able to put it in the form of a rst order7 autonomous system 0 Given a multi step map7 be able to write it as a system of l step maps Be able to provide examples of various types of dynamical sys tems eg maps and ODES7 linear and nonlinear7 autonomous and nonautonomous7 high order scalar equations and systems of equations 2 Well posedness Be able to state the existence and uniqueness theorem Be able to state the theorems given in class regarding continuity in initial conditions and parameters Be able to explain why these ideas are important in modeling physical systems Know the di erence between local and global existence 0 Given an explicit fx7 if7 A be able to determine a range of initial conditions where the initial value problem lVP x fX7t7 A is locally well posed Be able to determine a range of initial condi tions where the corresponding solutions exist for all time Be able to give examples of lVPs where uniqueness fails Be able to give examples of lVPs whose solutions blow up in nite time 3 Flows on the real line 0 Given i f either explicitly or as a graph7 be able to determine the equilibria of the ow and their stability by graphing Be able to determine the curvature of solutions by computing i f mi Given a phase portrait be able to give an example of an lVP i f that is consistent with that phase portrait Give a collection of solutions t be able to give an example of an lVP i f that is consistent with those solutions Be able to determine the stability of an equilibrium point by lin earization From linearization be able to determine the asymp totic rate of decay to or growth from the equilibrium point 4 Flows on the circle 0 Be able to write down ODEs for a uniform and nonuniform oscil lator 0 Be able to describe physical problems that are modeled by a uni form and nonuniform oscillator 0 Be able to identify the equilibrium points and periodic orbits for a given nonuniform oscillator 6 f aw by graphing f aw for di erent ranges of parameters Be able to determine their stability by this analysis 0 Be able to describe how a nonlinear oscillator behaves as the sys tem approaches a parameter regime where equilibrium points ap pear ghosts and bottlenecks 5 Bifurcations 0 Given a rst order equation with a parameter be able to determine when if at all a bifurcation occurs 0 Know the basic characteristics of the saddle node transcritical and pitchfork bifurcations including their bifurcation diagrams 0 Be able to determine if one of these bifurcations occurs in a given system that is not in the canonical form 6 One dimensional maps 0 Be able to nd xed points of simple maps explicitly and deter mine their stability using linear information Be able to identify xed points of more complicated maps graph ically and determine their stability Know the equation satis ed by a period m orbit Given the graph of the mth iteration map7 be able to nd period m solutions Know the stability criterion for period m orbits and be able to apply it Be able to describe in words the period doubling route to chaos Be able to describe how the Lyapunov exponent can indicate a systems tendency towards chaos You need not memorize the de nition of the Lyapunov exponent it Will be given if necessary Given the graph of a map7s Lyapunov exponent versus a parame ter7 be able to indicate where the onset of chaos is expected and where intermittancy is expected 7 Planar linear systems and linearization of planar nonlinear systems Be able to reproduce Figure 528 on page 137 of Strogatz Under stand how the eigenvalues and eigenvectors of a matrix determine the local phase portrait of the corresponding dynamical system Know the di erence between Lyapunov stability and asymptotic stability Be able to provide an example of an equilibrium point that is Lyapunov stable but not asymptotically stable7 and vice versa Know the di erence between attracting and Lyapunov stable Be able to provide and example of a xed point that is attracting but not Lyapunov stable Know that Lyapunov stable and attracting together imply asymptotically stable Know when the information from the linearization will predict the correct stability type of an equilibrium point under small nonlinear perturbations Know when the information from the linearization will predict the correct structure type of an equilibrium point under small nonlinear perturbations 0 Be able use information from the linearization to draw a rough picture of the local phase portrait for the corresponding nonlinear system 8 Periodic orbits in planar systems Know the two types of periodic solutions in a planar nonlinear system Be able to give an example of each Be able to state the Poincare Bendixson theorem and know how to apply it Know the implications of this theorem in terms of chaotic behavior 0 Given a nearly decoupled system in polar coordinates7 be able to design a trapping region for the corresponding ow 0 For more complicated ows7 be able to show that a given region is a trapping region 0 Know what nullclines are and how they are useful in determining the global ow Know what a Poincare map is and how it can be used to determine the existence and stability of a perioidic solution 0 Know the basic characteristics of a Hopf bifurcation 9 Hamiltonian systems 0 Be able to give examples of Hamiltonian system of ODEs 0 Given a smooth7 scalar valued function of two variables7 be able to write down the corresponding Hamiltonian system of di erential equations Given a system of di erential equations7 be able to determine if it is a Hamiltonian system and if so7 be able to nd the Hamiltonian Be able to show that the Hamiltonian is a conserved quantity Be able to explain why the level curves of the Hamiltonian are tangent to the trajectories of solutions in the phase plane Be able to draw the phase portrait using the level curves for the Hamiltonian 0 Be able to explain why isolated stable xed points cannot be asymptotically stable in a Hamiltonian system 0 Be able to show that the linearization about a xed point in a Hamiltonian system yields only saddles and centers Understand how this linear information is related to the existence of special solutions eg periodic solutions7 homoclinic orbits7 heteroclinic orbits in this type of system 10 Gradient systems Be able to give examples of gradient system of ODEs Given a potential function7 be able to write down the correspond ing gradient system Given a system of ODEs7 be able to determine whether or not it is a gradient system and if so7 be able to determine its potential function Be able to explain why the level curves of the potential function are perpendicular to the ow of solution trajectories Be able to explain why gradient systems cannot have nontrivial periodic solutions Be able to explain why gradient systems cannot have stable xed points that are not attracting Be able to explain why linearization of a gradient system will not predict spirals or centers 11 Gradient like systems 0 Know what a Lyapunov function is and why it is useful 0 Given a mechanical system with friction7 be able to come up with candidates for a Lyapunov function 0 Be able to explain how the existence of a Lyapunov function can help rule out periodic solutions in a region of the phase plane 0 Be able to explain how the existence of a Lyapunov function can help rule out xed points that are stable but not attracting Be able to provide an example of this phenomenon 12 Chaos in 3 D continuous systems Be able to show that solutions of the Lorenz system is invariant under the transformation x a x y a 7y Know what implica tions this has for the ow Be able to show that the Lorenz system is volume contracting Know what implications this has for the possible asymptotic be havior of solutions Understand how to use a Lyapunov function to show that the xed point at the origin is globally asymptotically stable for r lt 1 Be able to give a precise de nition of chaos Be able to give a precise de nition of an attractor Using these de nitions7 be able to explain why the Lorenz system exhibits chaos and possesses a strange attractor Be able to explain how the behavior of iterations for the Lorenz map gives evidence of chaos in the full system Be able to reproduce the bifurcation diagram for the Lorenz sys tem Strogatz7 p 3307 g 951

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