This was my slide presentation for my PhD defence. The topic is Hopf bifurcations and nonlinear dynamics in systems of PDEs.

1. ON THE DYNAMICAL CONSEQUENCES OF
GENERALIZED AND DEGENERATE
HOPF BIFURCATIONS: THIRD-ORDER EFFECTS
Todd Blanton Smith
PhD, Mathematics
University of Central Florida
Orlando, Florida
Spring 2011
Advisor: Dr. Roy S. Choudhury

2. Presentation Outline
• Intro to phase plane analysis and bifurcations
• Demonstration of analytical technique for dividing parameter
space into regions that support certain nonlinear dynamics.
• Three-mode laser diode system and predator-prey system
• Generalized Hopf bifurcations
• Four-mode population system
• Double Hopf Bifurcations

3. Phase plane analysis of nonlinear autonomous systems
Consider a general nonlinear
autonomous system
Fixed points satisfy
.
Add a small perturbation
A Taylor expansion gives behavior near the fixed point.
0

4. Bifurcations
• A bifurcation is a qualitative change in the phase
portrait of a system of ODE's.
A static bifurcation:
Eigenvalues are

5. Supercritical Hopf Bifurcations
They occur when the real part of a complex
conjugate pair of eigenvalues moves
through zero.

6. Subcritical Hopf Bifurcations
Two unstable fixed points collapse
onto a stable fixed point and cause
it to loose stability

7. Generalized Hopf Bifurcations
• A generalized Hopf bifurcation occurs in a
system with three dimensions when there is
one complex conjugate pair of eigenvalues and
the other eigenvalue is zero.
• In four dimensions there are three scenarios:
(Double Hopf)

8. Laser Diode and Predator-Prey systems
First we find generalized Hopf bifurcations and
nearby nonlinear dynamics in two systems:
Laser Diode Predator Prey

9. Locating a Generalized Hopf Bifurcation
In the Predator Prey model, our Jacobian matrix is
The eigenvalues satisfy the characteristic equation
where depend on the system parameters.
We insist on the form
which imposes the conditions and .

10. Analytical Construction of Periodic Orbits
- Use multiple scale expansions to construct periodic
orbits near the bifurcation.

11. Using these expansions and equating powers of
yields equations at
Where are differential operators and
are source terms.
These three equations can be solved simultaneously to
form a composite operator

12. The Source Terms
The first order sources are zero,
so we can guess a functional
form for the first order solution.

13. First-Order Solution
where we impose and .
This can be plugged into the source of the next order
composite equation
Now eliminate source terms that satisfy the
homogeneous equations.

14. Set the coefficients of equal to 0 to get
Change to polar coordinates for periodic orbits.

15. After the change of variables, setting equal the real and
imaginary parts gives the polar normal form.
The fixed points are

16. The post-generalized Hopf periodic orbit is
The evolution and stability of the orbit is determined by
the stability of the fixed point, so we evaluate the
Jacobian (eigenvalue) at the fixed point to get

17. Predator-Prey, parameter set 1

18. Predator-Prey, parameter set 2

19. Numerical confirmation of chaotic behavior
Autocorrelation function Power Spectral Density
Autocorrelation function PSD shows contributions
Approaches zero in a finite From many small frequencies
time

20. A stable quasi-periodic orbit is created after
the bifurcation in the laser diode system.

21. Periodic and Quasiperiodic Wavetrains from
Double Hopf Bifurcations in Predator-Prey
Systems with General Nonlinearities
• Morphogenesis- from Developmental Biology -
The emergence of spatial form and pattern from
a homogeneous state.

22. Turing studied reaction diffusion equations of the form
The reaction functions (or kinematic terms) R1 and R2 were
polynomials

23. The general two species predator-prey model
rate of predation per predator Prey birth rate
Carrying capacity
rate of the prey’s contribution to the predator growth
where N(t) and P(t) are the prey and predator populations, respectively

24. Here we investigate traveling spatial wave patterns in the form
where is the traveling wave, or “spatial”
variable, and v is the translation or wave speed.

25. The four-mode population system studied by Stefan Mancas:

26. Linear stability analysis
The fixed points of the system are
The functions F(N) and G(P) are kept general
during the analysis but are subsequently
chosen to correspond to three systems
analyzed by Mancas [28].

27. The Jacobian matrix
at the fixed point
(N0, M0, P0, Q0) is

28. The eigenvalues λ of the Jacobian satisfy the
characteristic equation
where bi, i = 1,…, 4 are given by

29. Double Hopf bifurcation- two pairs of purely imaginary
complex conjugates.
Hence we impose the characteristic form
This determines that the conditions for a double Hopf
bifurcation are

30. Analytical construction of periodic orbits
We use the method of multiple scales to construct
analytical approximations for the periodic orbits.
is a small positive non-dimensional
parameter that distinguishes
different time scales.
The time derivative becomes
Control Parameters:

31. where the Li, i = 1, 2, 3, 4, are the differential
operators

32. To find the solutions, we note that S1,i = 0 for i = 1, 2, 3, 4
and so we choose a first order solution

33. Suppressing secular source terms (solutions of the
homogeneous equations for i = 1) gives the requirement
Now we assume a second order solution of the form

34. This second order ansatz is plugged into the
composite operator with i = 2.
Evaluating the third order source term ᴦwith the first
3
and second order solutions and setting first order
harmonics equal to zero produces the normal form.
Note that c1 through c6 are complex. We can more
easily find periodic orbits by switching to polar
coordinates.

35. Plugging the polar coordinates
into the normal form and
separating real and imaginary
parts of the resulting
equations gives the normal
form in polar coordinates.

36. Periodic solutions are found by setting the two
radial equations equal to zero and solving for p1(T2)
and p2(T2).
There are four solutions: the initial equilibrium
solution, the Hopf bifurcation solution with frequency
ω1, the Hopf bifurcation solution with frequency ω2, and
the quasiperiodic solution with frequencies ω1 and ω2 .

37. The stability conditions for each of these solutions can be
determined with the Jacobian of the radial equations.
Evaluate this on the equilibrium solution and set the
eigenvalues negative to see the stability conditions:
and

38. Choosing μ1 = -0.24 and μ2 = 0.79
between L1 and L2 results in the stable
equilibrium solution shown in Figure 2

39. The second order deviation values μ1 = -0.4 and μ2 = 0.6 place
the sample point immediately after the line L1 where a Hopf
bifurcation occurs.
The values μ1 = -0.11 and μ2 = 0.9
place the sample point immediately
after L2.

40. Evaluating the Jacobian on the first Hopf
bifurcation solution yields the stability conditions:
When a20 < 0, the above conditions also allow the
second Hopf bifurcation solution to exist.

41. Evaluating the Jacobian on the second Hopf
bifurcation solution yields the stability conditions:
When b02 < 0, the above conditions also allow the
first Hopf bifurcation solution to exist.

42. μ1 = -0.06 , μ2 = 0.9
Quasiperiodic motion after line 3
Two frequency peaks
indicate quasiperiodic
motion.

43. The critical line L5, where a quasiperiodic
solution loses stability and may bifurcate into a
motion on a 3-D torus, is given here:

44. Immediately after the lines L4 and L5 the
solutions fly off to infinity in finite time.
The solutions in the regions after L4 and L5 are unstable

45. Conclusion
• In this dissertation we have demonstrated an analytical
technique for determining regions of parameter space that
yield various nonlinear dynamics due to generalized and
double-Hopf bifurcations.
• Thank You:
Dr Choudhury
Dr Schober
Dr Rollins
Dr Chatterjee
Dr Moore