ON THE DYNAMICAL CONSEQUENCES OF GENERALIZED AND DEGENERATE

HOPF BIFURCATIONS: THIRD-ORDER EFFECTS

Todd Blanton SmithPhD, Mathematics

University of Central FloridaOrlando, Florida

Spring 2011Advisor: Dr. Roy S. Choudhury

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

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

Bifurcations

• A bifurcation is a qualitative change in the phase portrait of a system of ODE's.

Eigenvalues are

A static bifurcation:

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

Supercritical Hopf Bifurcations

Subcritical Hopf Bifurcations

Two unstable fixed points collapseonto a stable fixed point and cause it to loose stability

 

 

• 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:

Generalized Hopf Bifurcations

(Double Hopf)

Laser Diode and Predator-Prey systems

First we find generalized Hopf bifurcations and nearby nonlinear dynamics in two systems:

Laser Diode Predator Prey

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 .

Analytical Construction of Periodic Orbits

- Use multiple scale expansions to construct periodic orbits near the bifurcation.

yields equations at

Where are differential operators and

are source terms.

Using these expansions and equating powers of

These three equations can be solved simultaneously to form a composite operator

The Source Terms

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

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.

Set the coefficients of equal to 0 to get

Change to polar coordinates for periodic orbits.

After the change of variables, setting equal the real and imaginary parts gives the polar normal form.

The fixed points are

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

Predator-Prey, parameter set 1

Predator-Prey, parameter set 2

Autocorrelation function Power Spectral Density

Numerical confirmation of chaotic behavior

Autocorrelation functionApproaches zero in a finitetime

PSD shows contributionsFrom many small frequencies

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

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.

Turing studied reaction diffusion equations of the form

The reaction functions (or kinematic terms) R1 and R2 were polynomials

The general two species predator-prey model

where N(t) and P(t) are the prey and predator populations, respectively

Prey birth rate

Carrying capacity

rate of the prey’s contribution to the predator growth

rate of predation per predator

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.

The four-mode population system studied by Stefan Mancas:

Linear stability analysisThe fixed points of the system

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].

are

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

The eigenvalues λ of the Jacobian satisfy the characteristic equation

where bi, i = 1,…, 4 are given by

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

Analytical construction of periodic orbits

We use the method of multiple scales to construct analytical approximations for the periodic orbits.

The time derivative becomes

is a small positive non-dimensional parameter that distinguishes different time scales.

Control Parameters:

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

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

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

This second order ansatz is plugged into the composite operator with i = 2.

Evaluating the third order source term ᴦ3 with the first 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.

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.

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 .

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

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

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.

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.

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.

μ1 = -0.06 , μ2 = 0.9

Quasiperiodic motion after line 3

Two frequency peaks indicate quasiperiodic motion.

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

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

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 ChoudhuryDr SchoberDr RollinsDr Chatterjee Dr Moore