Example 5: Consider x = px - y - x (x2 + y2) y = x + py - y(x + y2

There is an equilibrium point at the origin and the linearized system is

The eigenvalues are p ± i, so we expect a bifurcation when p = 0.

To see what happens as p passes through 0, we change to polar coordinates. The system becomes

Figure 7 Hopf bifurcation of system x = /x—y—x(x2+y2), y=x+^y-y(x2+y2). From Kuznetrov YA (2006) Andronov-Hopf bifurcation. Scholarpedia 1(10): 1858. http://www.scholarpedia.org/ article/Andronov-Hopf-bifurcation (accessed April 2008).

Figure 7 Hopf bifurcation of system x = /x—y—x(x2+y2), y=x+^y-y(x2+y2). From Kuznetrov YA (2006) Andronov-Hopf bifurcation. Scholarpedia 1(10): 1858. http://www.scholarpedia.org/ article/Andronov-Hopf-bifurcation (accessed April 2008).

Note that the origin is the only equilibrium point to this system, since 6f = 0. For / <0 the origin is a sink since /r—r3 < 0 for all r > 0. Thus, all solutions tend to the origin in this case. When / >0, the equilibrium becomes a source. So what else happens.? When / > 0 we have r = 0 if r = s//. So the circle of radius ^f/ is a periodic solution with period 2^. We also have r' > 0 if 0 < r < yf/, while r^ < 0 if r > yf/. Thus, all nonzero solutions spiral toward this circular solution as t n (see Figure 7).

This type of bifurcation is called Hopf bifurcation (or oscillatory bifurcation, Andronov-Hopf bifurcation), that is, a bifurcation from a branch of equilibria to a branch of periodic oscillations. Thus at a Hopf bifurcation, no new equilibrium arises. Instead, a periodic solution is born at the equilibrium point as / passes through the bifurcation value. That is, the so-called Hopf bifurcation connects equilibrium with periodic motion. Near Hopf bifurcations there is locally only one periodic solution to each parameter /. That is, only one half-branch of periodic solutions comes out from the stationary branch. Hopf bifurcation is the door that opens from the small room of equilibrium to the large hall of periodic solutions, which in turn is just a small part of the realm of functions. On the other hand, a Hopf bifurcation is the birthplace of a branch of periodic orbits. Hopf bifurcations are often involved in the destabilization of steady states in ecological models. In this way, Hopf bifurcations are connected to many interesting ecological effects like the paradox of enrichment. Furthermore, Hopf bifurcations play a prominent role in the formation of higher co-dimension bifurcations. Although these bifurcations cannot be observed directly in nature, their presence in models proves the existence of certain global bifurcations and local bifurcations of cycles. Most importantly, the Hopf bifurcation can serve as an indicator of chaotic dynamics.

Bifurcation Analysis

Basic Principles

The analysis of nonlinear phenomena requires tools that provide quantitative results and theoretical knowledge of

Chaos

Figure 8 Bifurcation diagram of logistic model xn+1 = rxn(1 —xn).

Figure 8 Bifurcation diagram of logistic model xn+1 = rxn(1 —xn).

nonlinear behavior that allows one to interpret these quantitative results, the tools of which we know are analytical and numerical.

As we know it, the complicated nonlinear ecosystem cannot be solved explicitly. So we have to study the system numerically integratedly, that is, via numerical simulation, and research about the long-term behavior of the solution. That is to say, the solution to the system with initial conditions in the first quadrant is obtained numerically for a biologically feasible range of parametric values. The so-called numerical simulation is such a technique that it uses time-series figures, bifurcation diagrams, phase portraits, and so on to describe the dynamic behavior of the system.

The bifurcation diagram, a 'family portrait', is a visual summary of the succession of how the dynamic behavior changes as the parameter increases. The parameter values increase horizontally across the bifurcation diagram (Figure 8 and Figure 10). The time series, which displays fluctuations of the dynamic behavior through time, is a familiar representation of a dynamic system (Figure 9). Furthermore, a less intuitive, but powerful representation of a dynamic system is phase portraits (Figures 11-17). The axes of the phase space are defined by the state variables so that, at a given moment, the state of the system is represented by points in the phase space. The power of the phase space representation comes about, because, as long as the system's fluctuations are bounded, an infinitely long trajectory inhabits a finite region of the phase space. In a deterministic system, the trajectory will eventually conform to another geometric object in the phase space: the attractor. In the phase space, a trajectory that is sufficiently 'near' an attractor will move toward it, on average, and once it is on the attractor it will not spontaneously move away.

Generally, the basis of numerical simulation is an iteration technique. The common algorithm for iteration is one of Gear single-step extrapolation method, Taylor series method, modified extended backward difference equation implicit method, forward Euler method, improved Euler method, Adams-Bashforth-Moulton method, Fehlberg x

60 80 100 120 140 160 180 t

60 80 100 120 140 160 180 t

60 80 100 120 140 160 180 3300 3350 3400 3450 3500 3550 3600 tt

Figure 9 Dynamical behaviors of the system [8] with T = 6, a1 = 1, a2 = 1.1, b1 = 4.1, b2 = 4.5, b3 = 0.6, d1 = 9, d2 = 10, c = 0.5, m = 0.2, a = 0.1, 3 = 0.15, <5-, = 0.2, <52 = 0.15, S3 = 0.0001, / = 0.5, the initial value (x(0), y(0), z(0)) = (3, 4, 5). (a) Time series of the prey population x when p >pmax = 53.87232 860, x(t)! 0, as t !+x>. (b) Time series of the prey population y when p >pmax, y(t)! 0, as t!+x>. (c) Time series of the predator population z evolving when p >pmax and (0, 0, z*(t)) is locally asymptotically stable.

fourth-fifth-order Runge-Kutta method, fourth-order classical Runge-Kutta method, etc. These methods use a fixed step size, providing no error estimation or correction. In fact, fourth-order classical Runge-Kutta method is one of practical and efficient algorithm. For more details, see Numerical Methods for Local Models.

The bifurcations then begin again with period 6, 12, 24,..., then back to chaos. In fact, it was discovered in Li-Yorke's famous paper 'Period three implies chaos' that any sequence with a period of three will display regular cycles of every other period as well as completely chaotic cycles.

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