low-dimensional hamiltonian dynamics in nonlinear optical ......state and the transfer of its...

Low-Dimensional Hamiltonian Dynamics In Nonlinear Optical

StructuresRoy Goodman, New Jersey Institute of Technology

Supported byDMS-0807284

Thursday, November 17, 11

Physical Motivation:Light propagation in a

3-well waveguide

xWhy three wells? • Recent work on two-waveguide systems shows symmetry-

breaking bifurcations and associated Duffing-like dynamics.

• Three waveguides provide the simplest system in which Hopf bifurcations, which lead to complex dynamics, are possible.

• Obvious interest in many-waveguide systems. Useful to proceed: Simple Geometry→ Complex Geometry,

Simple Dynamics→ Complex DynamicsThursday, November 17, 11

Nonlinear Schrödinger/Gross-Pitaevskii

• Standard model in:

• optics (Kerr nonlinearity)

• and Bose-Einstein Condensates (opposite nonlinearity)

• My interest: Stability of standing waves of the form:

• Principal tool: derivation of a finite-dimensional approximation and dynamical systems/asymptotic analysis of that ODE

especially the dynamics near an instability.

� = �⇥2x

� + V (x)� � |�|2�

�(x, t) = �(x)e�i�t

R�(x)2dx = ��22 = N

Solutions: from simple to complex

• One (narrow) waveguide: One localized solution whose frequency increases with increasing amplitude

• 2 waveguides:

• Even-symmetric ground state

• odd-symmetric excited state

• Ground state loses stability in symmetry breaking bifurcation at critical amplitude (Kirr/Kevrekidis/Shlizerman/Weinstein `08)

• Wobbling dynamics near new asymmetric solution (Marzuola/Weinstein `10)

−1 00

568 KIRR, KEVREKIDIS, SHLIZERMAN, AND WEINSTEIN

!0.3 !0.28 !0.26 !0.24 !0.22 !0.2 !0.18 !0.16 !0.14 !0.120

!10 !5 0 5 10

Fig. 1. (a) Bifurcation diagram for bound state solutions !(x, t) = e!i!t!!(x) of NLS-GPequation (2.1), with double-well potential (6.1) and cubic nonlinearity. Double-well parameters ares = 1, L = 6. " denotes the (nonlinear) frequency and N = N [!!], the squared L2 norm of !!

(optical power or particle number). The first bifurcation is from the zero state at the ground stateenergy of the double well. This state is an even function of x (symmetric). A secondary bifurcation,to an asymmetric state, at N = Ncr, is marked by a (red) circle. For N < Ncr the symmetric state((blue) solid line) is nonlinearly dynamically stable. For N > Ncr the symmetric state is unstable((blue) dashed line). The stable asymmetric state, appearing for N > Ncr, is marked by a (red)solid line. The (unstable) odd in x (antisymmetric) state is marked by a (green) dashed-dotted line.(b) Bound state solutions !!(x) plotted for a level set of N [!!] = 0.5.

restricted to small norm. As we shall see, a bifurcation occurring at small norm canbe ensured, for example, by taking the distance between wells in the double well tobe su!ciently large.

In [14] the precise transition point to symmetry breaking, Ncr, of the groundstate and the transfer of its stability to an asymmetric ground state was considered(by geometric dynamical systems methods) in the exactly solvable NLS-GP, with adouble-well potential consisting of two Dirac delta functions separated by a distanceL. Additionally, the behavior of the function Ncr(L) was considered. Another solvablemodel was examined by numerical means in [23]. A study of dynamics for nonlineardouble wells appeared in [27].

We study Ncr(L) in general. The value at which symmetry breaking occurs,Ncr(L), is closely related to the spectral properties of the linearization of NLS-GP

J.L. MARZUOLA AND M.I. WEINSTEIN NLS / GP WITH DOUBLE WELL POTENTIALS

−50 −40 −30 −20 −10 0 10 20 30 40 500

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−0.3

−0.2

−0.1

Figure 4. A numerical plot of an oscillatory solution to (1.1) at times a, b, c, and drespectively from the corresponding phase plane diagram of the finite dimensional Hamil-tonian truncation.

Fj = !jF ,(2.3)

for j = 0, 1 where

2|c0|2"20 + 2|c1|2"2

1 + 2(c0c1 + c1c0)"0"1

20 + c2

1"21 + 2c0c1"0"1

+ [ c1"1 + c0"0] R2 + [2c0"0 + 2c1"1 ] |R|2 + |R|2R,(2.4)

Frequency

Solutions: from simple to complex

• One (narrow) waveguide: One localized solution whose frequency increases with increasing amplitude

• 2 waveguides:

• Even-symmetric ground state

• odd-symmetric excited state

• Ground state loses stability in symmetry breaking bifurcation at critical amplitude (Kirr/Kevrekidis/Shlizerman/Weinstein `08)

• Wobbling dynamics near new asymmetric solution (Marzuola/Weinstein `10)

−1 00

568 KIRR, KEVREKIDIS, SHLIZERMAN, AND WEINSTEIN

!0.3 !0.28 !0.26 !0.24 !0.22 !0.2 !0.18 !0.16 !0.14 !0.120

!10 !5 0 5 10

Fig. 1. (a) Bifurcation diagram for bound state solutions !(x, t) = e!i!t!!(x) of NLS-GPequation (2.1), with double-well potential (6.1) and cubic nonlinearity. Double-well parameters ares = 1, L = 6. " denotes the (nonlinear) frequency and N = N [!!], the squared L2 norm of !!

(optical power or particle number). The first bifurcation is from the zero state at the ground stateenergy of the double well. This state is an even function of x (symmetric). A secondary bifurcation,to an asymmetric state, at N = Ncr, is marked by a (red) circle. For N < Ncr the symmetric state((blue) solid line) is nonlinearly dynamically stable. For N > Ncr the symmetric state is unstable((blue) dashed line). The stable asymmetric state, appearing for N > Ncr, is marked by a (red)solid line. The (unstable) odd in x (antisymmetric) state is marked by a (green) dashed-dotted line.(b) Bound state solutions !!(x) plotted for a level set of N [!!] = 0.5.

restricted to small norm. As we shall see, a bifurcation occurring at small norm canbe ensured, for example, by taking the distance between wells in the double well tobe su!ciently large.

In [14] the precise transition point to symmetry breaking, Ncr, of the groundstate and the transfer of its stability to an asymmetric ground state was considered(by geometric dynamical systems methods) in the exactly solvable NLS-GP, with adouble-well potential consisting of two Dirac delta functions separated by a distanceL. Additionally, the behavior of the function Ncr(L) was considered. Another solvablemodel was examined by numerical means in [23]. A study of dynamics for nonlineardouble wells appeared in [27].

We study Ncr(L) in general. The value at which symmetry breaking occurs,Ncr(L), is closely related to the spectral properties of the linearization of NLS-GP

J.L. MARZUOLA AND M.I. WEINSTEIN NLS / GP WITH DOUBLE WELL POTENTIALS

−50 −40 −30 −20 −10 0 10 20 30 40 500

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−0.3

−0.2

−0.1

Figure 4. A numerical plot of an oscillatory solution to (1.1) at times a, b, c, and drespectively from the corresponding phase plane diagram of the finite dimensional Hamil-tonian truncation.

Fj = !jF ,(2.3)

for j = 0, 1 where

2|c0|2"20 + 2|c1|2"2

1 + 2(c0c1 + c1c0)"0"1

20 + c2

1"21 + 2c0c1"0"1

+ [ c1"1 + c0"0] R2 + [2c0"0 + 2c1"1 ] |R|2 + |R|2R,(2.4)

Frequency

3 waveguides: what is known• At zero amplitude: 3 standing

• 2 even-symmetric modes

• 1 odd-symmetric mode

• Standing waves may lose stability as amplitude increased, new asymmetric modes appear Kapitula, Kevrekidis, Chen ’06

• Question remains: what are the dynamics? What other types of solution may arise?

Frequency

−18 −17 −16 −15 −14 −13 −12 −11 −10 −90

Odd solutionsEven SolutionsAsymmetric Solutions

Types of questions we can ask

• There are known to br several families of standing wave solutions

• Longstanding attention: stability (spectral, nonlinear, etc.) of these solutions

• More recent attention: the dynamics that arise due to instabilities

• What other types of simple solutions exist?

�(x, t) = �(x)e�i⌦t

Mathematical Plan of Talk

•Discuss mathematical assumptions

•Derive a finite dimensional (ODE) system that captures the dynamics

• Linear stability/bifurcation analysis of a simple solution

•Use Hamiltonian averaging to further reduce the dynamics and explain simulations

•Numerical simulation of dynamics in stable and unstable regimes

W + s � W − s �

�1�2 �3

• The potential satisfies & is even:

• The operator has exactly 3 discrete eigenvalues

• The eigenvalues are nearly resonant

• More specifically

Assumptions on potentialDefine the operator , then H = ��2

+ V (x)

V (x) < 0V (�x) = V (x)

�1 < �2 < �3 < 0

�3 � 2�2 + �1 ⌧ 1

�2 � �1 = W + �

�3 � �2 = W � �

W > 0, � ⌧ 1

W + s � W − s �

�1�2 �3

• The potential satisfies & is even:

• The operator has exactly 3 discrete eigenvalues

• The eigenvalues are nearly resonant

• More specifically

Assumptions on potentialDefine the operator , then H = ��2

+ V (x)

V (x) < 0V (�x) = V (x)

�1 < �2 < �3 < 0

�3 � 2�2 + �1 ⌧ 1

�2 � �1 = W + �

�3 � �2 = W � �

W > 0, � ⌧ 1

can be constructed via inverse scattering.V (x)Thursday, November 17, 11

−4 −3 −2 −1 0 1 2 3 4−20

−4 −2 0 2 4−1

−0.5

1�1(x), �1=−11

−4 −2 0 2 4−1

−0.5

1�2(x), �2=−10.1

−4 −2 0 2 4−1

−0.5

1�3(x), �3=−9

Example potentialSimilar to Kapitula/Keverekidis/Chen (2006) triple well

but inverse scattering construction gives more control over the spectrum, less degenerate eigenvalues

V (x) = V0(x+ L) + V0(x) + V0(X � L)

−4 −3 −2 −1 0 1 2 3 4−20

−4 −2 0 2 4−1

−0.5

1�1(x), �1=−11

−4 −2 0 2 4−1

−0.5

1�2(x), �2=−10.1

−4 −2 0 2 4−1

−0.5

1�3(x), �3=−9

Example potentialSimilar to Kapitula/Keverekidis/Chen (2006) triple well

but inverse scattering construction gives more control over the spectrum, less degenerate eigenvalues

V (x) = V0(x+ L) + V0(x) + V0(X � L)

Stability of this “odd” mode the

topic of this talkThursday, November 17, 11

•When , solutions are simply the eigenvalue/eigenfunction pairs

•When , solutions persist while maintaining the symmetry of the eigenfunction while

• Very little dependence of shape on , so we can use linear eigenfunctions as a basis

Nonlinear standing modes

N �= 0� = �j(N )

−5 0 5

−0.25

−0.2

−0.15

−0.1

−0.05

−5 0 5

−0.1

−5 0 5

−0.1−0.05

00.050.10.15

⇥�(x) = ��2x

�+ V (x)�� |�|2�⇥�⇥2 = N

The Hamiltonian Hopf bifurcationStability lost when two pairs of imaginary eigenvalues of opposite Krein signature collide and split into a quartet

of nontrivially complex eigenvalues

Re �

Im �

Re �

Im �

This bifurcation is not associated with the creation of new fixed points, but with new periodic orbits. Structure depends on nonlinear terms in

the Normal Form.Thursday, November 17, 11

The present system

● Linearizing about the mode at by letting

yields the eigenvalues

⇥(x, t) = (�2(x) + �(x)e�t)e�i�2t

� = ±i(�1,3 � �2)

The present system

⇥(x, t) = (�2(x) + �(x)e�t)e�i�2t

� = ±i(�1,3 � �2)

● Our assumptions give rise to nearly double eigenvalues on the imaginary axis and the correct Krein signatures for instability

⇥ = ±i(W ± �)

The present system

⇥(x, t) = (�2(x) + �(x)e�t)e�i�2t

� = ±i(�1,3 � �2)

● Our assumptions give rise to nearly double eigenvalues on the imaginary axis and the correct Krein signatures for instability

⇥ = ±i(W ± �)

● We anticipate (and show) that the HH bifurcation should take place at .N / �

Finite-dimensional reductionDecompose the solution as

⇥ = c1(t)�1(x) + c2(t)�2(x) + c3�3(x) + �(x; t)

projection onto eigenmodes

⇥ = c1(t)�1(x) + c2(t)�2(x) + c3�3(x) + �(x; t)

projection onto eigenmodes �(x, t) ? �j(x)

⇥ = c1(t)�1(x) + c2(t)�2(x) + c3�3(x) + �(x; t)

dt� �

+ a1111

+ a1113

+ 2 |c1

) + a1122

|2 + c1

+a1133

|2 + c1

) + a1223

+ 2 |c2

) + a1333

, �)

dt� �

+ a1122

+ 2 |c1

) + 2a1223

+a2222

+ a2233

|2 + c2

) = R2

, �)

dt� �

+ a1113

+ a1133

+ 2 |c1

) + a1223

|2 + c1

+a1333

|2 + c1

) + a2233

+ 2 |c2

) + a3333

, �)

i⇥t� �H� + |�|2� = Rcont

, �);

Whereajklm =

R�j(x)�k(x)�l(x)�m(x) dx

projection onto eigenmodes �(x, t) ? �j(x)

Finite Dimensional Reduction 2Ignore the effect of (justify yourself later). Get a Hamiltonian system:

�(x, t)

H =�1 |c1|2 + �2 |c2|2 + �3 |c3|2 � 12a1111 |c1|

4 � a1113 |c1|2 (c1c3 + c1c3)

� a1122

⇣12c

22 + 2 |c1|2 |c2|2 + 1

2 c21c

⌘� a1133

⇣12c

23 + 2 |c1|2 |c3|2 + 1

2 c21c

� a1223

⇣2 |c2|2 (c1c3 + c1c3) + c1c

22c3 + c1c

⌘� a1333 |c3|2 (c1c3 + c1c3)

� 12a2222 |c2|

4 � a2233

⇣12c

23 + 2 |c2|2 |c3|2 + 1

2 c22c

⌘� 1

2a3333 |c3|4.

Note: the conjugate pairs are (qj , pj) = (cj , icj)

icj =dH

dcjSo that

Further reductionMaking the change of variables

The Hamiltonian is θ-independent (from L2 conservation) so we can eliminate θandρto get the following 2 d.o.f. Hamiltonian

H =(�1 � �2)|z1|2 + (�3 � �2)|z3|2 �1

2a1111|z1|4 � a1113|z1|2(z1z3 + z1z3)

� a11222

(N � |z1|2 � |z3|2)(z21 + 4|z1|2 + z21)�a11332

(z21 z23 + 4|z1|2|z3|2 + z21z

� a1223(N � |z1|2 � |z3|2)(z1z3 + 2z1z3 + 2z1z3 + z1z3)� a1333|z3|2(z1z3 + z1z3)

2a2222(N � |z1|2 � |z3|2)2 �

a22332

(N � |z1|2 � |z3|2)(z23 + 4|z3|2 + z23)

2a3333|z3|4.

c1(t) = z1(t)ei�(t); c2(t) = �(t)ei�(t); c3(t) = z3(t)e

i�(t).

We are interested in the stability of, and dynamics near, the trivial solution

Detecting the Hopf BifurcationLinearize and split into real and imaginary parts

giving the symplectic linear ODE

✓z1(t)z3(t)

✓u1(t) + iv1(t)u3(t) + iv3(t)

✓0 �M�

✓W � �+ (m±a1122 � a2222)N m±a1223N

m±a1223N �W � �+ (a2233 +m±a2222)N

Detecting the Hopf BifurcationLinearize and split into real and imaginary parts

giving the symplectic linear ODE

✓z1(t)z3(t)

✓u1(t) + iv1(t)u3(t) + iv3(t)

✓0 �M�

✓W � �+ (m±a1122 � a2222)N m±a1223N

m±a1223N �W � �+ (a2233 +m±a2222)N

Hamiltonian Hopf bifurcation where this matrix has a double eigenvalue, i. e. where its characteristic polynomial

has a double root.Thursday, November 17, 11

The characteristic polynomial has discriminant:

Detecting the Hopf Bifurcation 2

d4(�)N4 + d3(�)N

3 + d2(�)N2 + d1(�)N + d0(�) = 0

d4(�)N4 + d3(�)N

3 + d2(�)N2 + d1(�)N + d0(�) = 0

Solve for the amplitude by perturbation seriesNHH =

�a1122 ± a1223 + a2222 � a2233+O(�2).

d4(�)N4 + d3(�)N

3 + d2(�)N2 + d1(�)N + d0(�) = 0

�a1122 ± a1223 + a2222 � a2233+O(�2).

Odd solution unstable--discriminant

Odd solution stable--discriminant

NHH−1 −0.5 00

Dependence of critical amplitude on ε

d4(�)N4 + d3(�)N

3 + d2(�)N2 + d1(�)N + d0(�) = 0

�a1122 ± a1223 + a2222 � a2233+O(�2).

Odd solution unstable--discriminant

Odd solution stable--discriminant

NHH−1 −0.5 00

Dependence of critical amplitude on ε

1 2 3 4 5−0.5

Real(h)

1 2 3 4 5

Imag(h)Numerically

calculated Spectrum

● Assume small nonlinearity

Analysis of ResonanceN = �⇥

H = W⇤1 �W⇤3 + �H1(⇤1, ⇤3, ⇥1, ⇥3)

● Put variables in canonical polar coordinates, giving Hamiltonian

Analysis of Resonance

The so-called 1:-1 resonance

N = �⇥

H = W⇤1 �W⇤3 + �H1(⇤1, ⇤3, ⇥1, ⇥3)

Analysis of Resonance

Symplectic change of variables transforms the system to●

H = �WJ3 + �H1(J1, J3,⇥1,⇥3)

The so-called 1:-1 resonance

N = �⇥

H = W⇤1 �W⇤3 + �H1(⇤1, ⇤3, ⇥1, ⇥3)

Reduction to 1.5 dimensions via averagingH = �WJ3 + �H1(J1, J3,⇥1,⇥3)

Define a new time variable⌧ = 3

Hreduced = � �

WH1(J1,⇤1,�h)� �

WH(J1,⇤1,�h; ⇥)

The Hamiltonian becomes●

Periodic in , may be pushed off to by averaging⌧ O(�2)

Hreduced = � �

WH1(J1,⇤1,�h)� �

WH(J1,⇤1,�h; ⇥)

Periodic in , may be pushed off to by averaging⌧ O(�2)

Geometrical structure of dynamics determined by average term Haverage = � �

WH1(J1,⇥1,�h)

Hreduced = � �

WH1(J1,⇤1,�h)� �

WH(J1,⇤1,�h; ⇥)

The averaged system

• nonlinearity strength

• conserved quantity

˜H1 = �1(⇥)J1 + �2(⇥)J21 + �3(⇥)

pJ1 � h (�h+ 2J1 � 1) cos⇤1

This system depend on two parameters:⇥ = N/�

h = J3 = |z3|2 � |z1|2

The averaged system

˜H1 = �1(⇥)J1 + �2(⇥)J21 + �3(⇥)

pJ1 � h (�h+ 2J1 � 1) cos⇤1

h = J3 = |z3|2 � |z1|2

The system with is especially important because it contains the trivial solution whose stability we’re considering

z1 = z3 = 0

The averaged system

˜H1 = �1(⇥)J1 + �2(⇥)J21 + �3(⇥)

pJ1 � h (�h+ 2J1 � 1) cos⇤1

We take the nonlinearity strength as the bifurcation parameter.

h = J3 = |z3|2 � |z1|2

The system with is especially important because it contains the trivial solution whose stability we’re considering

z1 = z3 = 0

0 0.1 0.2 0.3 0.4 0.5−3

Phase portraits of averaged systemNew fixed point and homoclinic orbit appear at the threshold nonlinearity strength

0 0.1 0.2 0.3 0.4 0.5−3

�HH = ± 1

±a1223 + a1122 � a2222 + a2233� < �HH � > �HH

0 0.1 0.2 0.3 0.4 0.5−3

Phase portraits of averaged systemNew fixed point and homoclinic orbit appear at the threshold nonlinearity strength

0 0.1 0.2 0.3 0.4 0.5−3

�HH = ± 1

±a1223 + a1122 � a2222 + a2233� < �HH � > �HH

This is exactly the critical amplitude for the HH bifurcation found by perturbation theory!

1 2 3 4 5−0.5

Real(h)

ODE & PDEsimulations ●

Real(z1) Poincaré Section

0 75 150!1

(a), N=0.35

|�(t)|

−2 0 2 4 6 8x 10−4

5x 10−4

1 2 3 4 5−0.5

Real(h)

Real(z1) Poincaré Section |�(t)|

0 200 400−0.5

(c), N=1.0

0 0.2 0.4−0.2

−0.1

1 2 3 4 5−0.5

Real(h)

Real(z1) Poincaré Section |�(t)|

0 40 80

−0.5

(d), N=3.0

−0.5 0 0.5

−0.5

Analysis of supercritical dynamicsNext goal: Understand the nonlinear dynamics and the transition from regular to complex dynamics

Classifying the bifurcation: Different types of HH bifurcation depending on nonlinearity (Chow & Kim). Here, a homoclinic structure & a new periodic orbit appears at distance from the origin.

0.05 0.1 0.15 0.2 0.25 0.3

Re(!3)

N �NHH

0.05 0.1 0.15 0.2 0.25 0.3

Re(!3)

N �NHH

StationaryOdd Solution

0.05 0.1 0.15 0.2 0.25 0.3

Re(!3)

N �NHH

StationaryOdd Solution

AsymmetricPeriodic Orbit

Existence of Periodic OrbitsTheorem of Chow & Kim ’88

Writing where is in normal form

H = H2 + ✏H2 + H4

H2 = |z1|2 � |z3|2with , and resonant, and

non-resonant, then there are periodic orbits of the form

⇠(t) = eJAta

where is an extremum of subject to the constraint .

Enumerating the Periodic Orbits

�⌥0 +⌥1⇢

21 +⌥2⇢

�⇢1⇢3 + ⇤

�N(⇢21 + ⇢23)� (⇢41 + 6⇢21⇢

23 + ⇢43)

�cos⇥ = 0

�⇢21 + ⇢23 �N

�⇢1 sin⇥ = 0

�⇢21 + ⇢23 �N

�⇢3 sin⇥ = 0

Solve a Lagrange multiplier in polar coordinates

⇥ = ✓1 + ✓3

�⌥0 +⌥1⇢

21 +⌥2⇢

�⇢1⇢3 + ⇤

�N(⇢21 + ⇢23)� (⇢41 + 6⇢21⇢

23 + ⇢43)

�cos⇥ = 0

�⇢21 + ⇢23 �N

�⇢1 sin⇥ = 0

�⇢21 + ⇢23 �N

�⇢3 sin⇥ = 0

⇥ = ✓1 + ✓3

The 3 deformed eigenfunctions(relative fixed points)

�⌥0 +⌥1⇢

21 +⌥2⇢

�⇢1⇢3 + ⇤

�N(⇢21 + ⇢23)� (⇢41 + 6⇢21⇢

23 + ⇢43)

�cos⇥ = 0

�⇢21 + ⇢23 �N

�⇢1 sin⇥ = 0

�⇢21 + ⇢23 �N

�⇢3 sin⇥ = 0

⇥ = ✓1 + ✓3

2 families of Previously undiscovered relative periodic orbits

�⌥0 +⌥1⇢

21 +⌥2⇢

�⇢1⇢3 + ⇤

�N(⇢21 + ⇢23)� (⇢41 + 6⇢21⇢

23 + ⇢43)

�cos⇥ = 0

�⇢21 + ⇢23 �N

�⇢1 sin⇥ = 0

�⇢21 + ⇢23 �N

�⇢3 sin⇥ = 0

⇥ = ✓1 + ✓3

Increasing N

�⌥0 +⌥1⇢

21 +⌥2⇢

�⇢1⇢3 + ⇤

�N(⇢21 + ⇢23)� (⇢41 + 6⇢21⇢

23 + ⇢43)

�cos⇥ = 0

�⇢21 + ⇢23 �N

�⇢1 sin⇥ = 0

�⇢21 + ⇢23 �N

�⇢3 sin⇥ = 0

⇥ = ✓1 + ✓3

Increasing N

New solutions bifurcate on or near even subspace

�⌥0 +⌥1⇢

21 +⌥2⇢

�⇢1⇢3 + ⇤

�N(⇢21 + ⇢23)� (⇢41 + 6⇢21⇢

23 + ⇢43)

�cos⇥ = 0

�⇢21 + ⇢23 �N

�⇢1 sin⇥ = 0

�⇢21 + ⇢23 �N

�⇢3 sin⇥ = 0

⇥ = ✓1 + ✓3

Increasing N

�⌥0 +⌥1⇢

21 +⌥2⇢

�⇢1⇢3 + ⇤

�N(⇢21 + ⇢23)� (⇢41 + 6⇢21⇢

23 + ⇢43)

�cos⇥ = 0

�⇢21 + ⇢23 �N

�⇢1 sin⇥ = 0

�⇢21 + ⇢23 �N

�⇢3 sin⇥ = 0

⇥ = ✓1 + ✓3

Increasing N

At Hamiltonian Hopf bifurcation, this branch detaches from origin

�⌥0 +⌥1⇢

21 +⌥2⇢

�⇢1⇢3 + ⇤

�N(⇢21 + ⇢23)� (⇢41 + 6⇢21⇢

23 + ⇢43)

�cos⇥ = 0

�⇢21 + ⇢23 �N

�⇢1 sin⇥ = 0

�⇢21 + ⇢23 �N

�⇢3 sin⇥ = 0

⇥ = ✓1 + ✓3

Mathematical Extensions

• Rigorous perturbation theory relating averaged system to full equations/demonstration of chaos

• Demonstrate that ODE solutions--new periodic orbits, relative periodic orbit, and chaotic dynamics--are shadowed by PDE solutions a la Marzuola & Weinstein, Pelinovsky & Phan

• In this system, slight supercriticality implies perturbations remain small. Increasing the number of degrees of freedom to four can lead to order-one excursions: Arnol’d diffusion

• Mathematical techniques developed for celestial mechanics. Can we design potentials for which the dynamics will mimic other phenomena seen in celestial mechanics? Can we do something useful with this?

Questions: experimental physics

• Laboratory verification of Hamiltonian Hopf bifurcation, appearance of complex dynamics

•Design/build more complex waveguides in which more complicated nonlinear dynamics occurs and (maybe) could be put to use.

Questions: experimental physics

• Laboratory verification of Hamiltonian Hopf bifurcation, appearance of complex dynamics

•Design/build more complex waveguides in which more complicated nonlinear dynamics occurs and (maybe) could be put to use.

Thanks!

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