New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

Nonlinear Dynamics

by: Kylie Bartoletti DVM

Nonlinear Dynamics PHYS 7224

Kylie Bartoletti DVM

GPA 3.67


Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

Class Notes
25 ?




Popular in Course

Popular in Physics 2

This 0 page Class Notes was uploaded by Kylie Bartoletti DVM on Monday November 2, 2015. The Class Notes belongs to PHYS 7224 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/234290/phys-7224-georgia-institute-of-technology-main-campus in Physics 2 at Georgia Institute of Technology - Main Campus.


Reviews for Nonlinear Dynamics


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 11/02/15
Example supercritical pitchfork bifurcation u k Normalform 3 L120 rlt0 uzru u gt u0iJ7 rgt0 k1 k Example subcritical pitchfork bifurcation u Normal form 3 u 0 ix r r lt 0 u ru u gt n u 0 r gt 0 X r Note There are other types of subcritical and supercritical bifurcations such as a Hopf bifurcation which occurs when two complex conjugate eigenvalues simultaneously become unstable The Hopf bifurcation leads to oscillatory nonlinear states 1 The normal forms characteristic of each bifurcation type represent a Taylor expansion of the evolution equation L2 f u r in terms of the distance to the bifurcation point in the space of parameters here r and variables here u The leading order term a constant always vanishes as the growth rate has to vanish at the bifurcation point u r 0 0 Higher order terms might vanish as well due to certain symmetries of the evolution equations e g pitchfork bifurcation has a u gt u symmetry Bifurcations Into Stripe States Confining the pattern forming system to a finite region with periodic boundary conditions reduces a problem with a continuum of unstable eigenvalues to one with discrete eigenvalues that can be treated within bifurcation theory However this is a degenerate bifurcation at which a finite number of eigenvalues go unstable together due to the rotational symmetries of the system Consider the example of a stationary transition to stripes We set up this problem in a geometry that is periodic over some long length l in the x direction which will be the direction normal to the stripes but periodic over a short distance in the ydirection so that the bifurcating solution is uniform in this direction From the parity symmetry x gt x it can be deduced that there will be two real eigenvalues that pass through zero together corresponding to wave vectors 6 and 61 giving the spatial dependences 6 From these we can form real solutions proportional to cosqx and sinqx or generally cosqx where is an arbitrary phase that represents the translational symmetry of the system the stripes may be laid down at any position along the x direction Alternatively we can write the solution in compleX notation u oc Aeiqx co Here A arequotj is a complex amplitude with the magnitude a IA giving the size of the disturbance and the phase giving the position of the stripes Thus the difference from the simple stationary bifurcation discussed previously e g a pitchfork bifurcation is that the new solution is represented by a complex rather than a real variable It is easy to see the implications of this change on the normal form Remember that changing the phase of the compleX amplitude gives a translation of the stripes through symmetryrelated states This means that the normal form must be invariant under a phase change of A ie the form of the equation must not change if we replace A gt A6 with 5 some constant Using the same idea of a Taylor eXpansion of the dynamics as for the simple bifurcations and incorporating this symmetry constraint we deduce that the normal form is atAeriA2A 1 This is analogous to the pitchfork bifurcation For the negative sign of the nonlinear term we have a supercritical bifurcation with new solutions A requot developing for small positive r with arbitrary These are easily seen to be stable within the dynamics described by E 1 For the positive sign in Eq 1 the bifurcation is subcritical and unstable solutions A rei eXist for small negative r To seek stable finite amplitude solutions we would have to investigate the right hand side of Eq 1 by extending the Taylor eXpansion to higher order 61A 2 rAA2A gA4A Again this is only a reliable predictor of the nonlinear solutions if g happens to be small 1 An important difference from the simple bifurcation is that the bifurcation to stripes is always of the pitchfork type as a consequence of translational symmetry The analogue to the transcritical bifurcation does not eXist in this case independent of whether the system shows a u gt u symmetry As we will see later the transition to a hexagonal state may be transcritical Nonlinear saturation in the SwiftHohenberg equation As we have learned previously the SwiftHohenberg equation describes a typeIS instability to a cellular structure in onedimension atu m l62u u3 As r is increased from negative values the uniform state u 0 develops an instability at r 0 to a mode with critical wave number qc 1 For small positive r we can find a perturbation growing from very small amplitude ux t oc 6 cos x We wish to understand the nonlinear saturation of this state for small r A first guess might be to look for a solution which is just a saturation of the growing linear mode ux a1 cos x with al to be determined To see if this ansatz is a solution we substitute it into the equation for stationary solutions atu 0 3 l 0 ra1 a13 cos x a13 cos 3x 4 4 where we have rewritten cos3 x generated by the L13 term as a sum of cos x and cos 3x terms Now since the Fourier modes cos nx are linearly independent the coefficient on the right hand side of each mode must be zero For instance the coefficient of cos x determines a1 ral ia1320 gt a1Oiq43r12 However equating the coefficient of cos 3x to zero selects the wrong solution 1 Zal30 gt all 0 This inconsistency shows that the original guess was incomplete We rapidly realize however that the problem is removed by adding a small amplitude of cos 3x to the original ansatz so that now ux a1 cos x a3 cos 3x With this new ansatz we obtain the following equality 3 3 3 l 3 3 0 m1 Za13 Za12a3 Eala32cos x m3 64a3 Za13 Ea12a3 Za cos 3x 3 3 3 l a1a32 a12a3 cos 5x ala32 cos 7x a33 cos 9x 4 4 4 4 where again setting the coefficients of different harmonics cos nx to zero determines a1 and a3 As we are looking for a solution close to the bifurcation point where the effect of nonlinear terms is small a3 will be small compared to al Therefore setting the coefficient of cos x to zero gives 3 3 3 3 ra1 a13 a12a3 Eala32 N ml Za13 0 4 4 ie the same equation for all as before Similarly the coefficient of cos 3x gives 3 l 3 3 l l m 64a a3 a2a a3z 64a a3 0 gt a a3ocr32 3 3 4 l l 3 3 3 l 3 l 2 4 4 256


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Steve Martinelli UC Los Angeles

"There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Anthony Lee UC Santa Barbara

"I bought an awesome study guide, which helped me get an A in my Math 34B class this quarter!"

Jim McGreen Ohio University

"Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.