Hopf bifurcation analysis and first Lyapunov coefficient (3)
Compute the first Lyapunov coefficient for the Brusselator system:
The equilibrium point:
Assume α>0 fixed and take β as a bifurcation parameter to show that the system exhibits a supercritical Hopf bifurcation at X0(α, β=β0), where β0=1+α2.
The Jacobian matrix and its transpose:
Analyse the local stability with the resource function BialternateSum, with β as control parameter:
The critical value of the Hopf bifurcation:
Then, the Brusselator system is locally asymptotically stable at X0(μ) for β<β0 and locally asymptotically unstable for β>β0 (appears a stable limit cycle surrounded the unstable equilibrium point). We verify the previous conclusion by the transversality condition:
The non-trivial equilibrium point at β=β0:
Shift the non-trivial equilibrium point to the origin:
Verify that indeed the non-trivial equilibrium was shifted to the origin:
The Jacobian matrix in the new state variables:
The linear approximation at the origin and its transpose at β=1+α2 and α=ω:
The critical eigenvectors q,
, p,
:
The normalization 〈pn,q〉=1
The third-order Taylor approximation for the polynomial system
Linear function:
Bilinear function:
Trilinear function:
Verify that the Taylor series expansion of the Brusselator system is correct:
The first Lyapunov coefficient is given by:
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/679648b8792ddc0d.png)
where
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/4cbc0964ed9dd85f.png)
and h20, h11 are given by
h20=(2 ⅈω 𝕀2×2-AA)-1BB(q,q), h11=-AA-1BB(q,q)
First, compute h20, h11 and C1 as follows:
Finally, compute the first Lyapunov coefficient:
The first Lyapunov coefficient is clearly negative for all positive α. Thus, the Hopf bifurcation is nondegenerate and always supercritical.
Compute the first Lyapunov coefficient for the following predator-prey system:
The orbitally equivalent polynomial system:
The Jacobian matrix for the orbitally equivalent polynomial system:
The non-trivial equilibrium point:
The linear approximation at X0(μ):
Analyse the local stability with the resource function BialternateSum, with α as control parameter:
The critical value of the Hopf bifurcation:
Then, the system is locally asymptotically stable for α>α0 and locally asymptotically unstable for α<α0 (with c>d), and can be confirmed by the transversality condition:
The non-trivial equilibrium point at α=α0:
Shift the non-trivial equilibrium point to the origin:
Verify that indeed the non-trivial equilibrium was shifted to the origin:
The Jacobian matrix in the new state variables:
The linear approximation at the origin and at
and
:
The critical eigenvectors q,
, p,
:
The normalization 〈pn,q〉=1
The third-order Taylor approximation for the polynomial system
Linear function:
Bilinear function:
Trilinear function:
Verify that the Taylor series expansion of the polynomial system is correct:
The first Lyapunov coefficient is given by:
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/5a2005a58ead676e.png)
where
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/4cbc0964ed9dd85f.png)
and h20, h11 are given by
h20=(2 ⅈω 𝕀2×2-AA)-1BB(q,q), h11=-AA-1BB(q,q)
First, compute h20, h11 and C1 as follows:
Finally, we compute the first Lyapunov coefficient:
Clearly, the first Lyapunov coefficient is strictly negative for all combinations of the fixed parameters. Therefore, a unique limit cycle emerges from the non-trivial equilibrium via Hopf bifurcation for α<α0.
Compute the first Lyapunov coefficient for the following nonlinear feedback-control system of Lur'e type:
For all values of (α>0,β>0), the Lur'e system has two equilibria, the origin and (1,0,0). At the origin, this system exhibits a supercritical Hopf bifurcation when α=α0, where
:
The Jacobian matrix and its transpose:
Analyse the local stability with the resource function BialternateSum, with β as control parameter:
The critical value of the Hopf bifurcation:
Then, the Lur'e system is locally asymptotically stable at X0(μ) for α>α0 and locally asymptotically unstable for α<α0 (appears a stable limit cycle surrounded the unstable equilibrium point). We verify the previous conclusion by the transversality condition:
The linear approximation at the origin and its transpose at
and β=ω2:
The critical eigenvectors q,
, p,
:
The normalization 〈pn,q〉=1
The third-order Taylor approximation for the polynomial system
Linear function:
Bilinear function:
Trilinear function:
We verify that the Taylor series expansion of the Lur'e system is correct:
The first Lyapunov coefficient is given by:
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/563e49ecd0c4608f.png)
where
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/4cbc0964ed9dd85f.png)
and h20, h11 are given by
h20=(2 ⅈω 𝕀3×3-AA)-1BB(q,q), h11=-AA-1BB(q,q)
First, compute h20, h11 and C1
Finally, compute the first Lyapunov coefficient:
The first Lyapunov coefficient is strictly negative for all positive β. Thus, the Hopf bifurcation is nondegenerate and always supercritical.
Neimark-Sacker bifurcation analysis and first Lyapunov coefficient (19)
The recurrence equation 𝓊k+1=𝓇 𝓊k+1(1-𝓊k-1) is simple population dynamical model, where 𝓊k depict for the density of a population at time k, and 𝓇 is a growth rate. Introducing 𝓊k=𝓊k-1, the above equation can be rewritten as
𝓊k+1=𝓇 𝓊k(1-𝓋k), 𝓋k+1= 𝓋k,
which, in turn, defines the following two-dimensional discrete-time dynamical system
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/0e5920627acaeb42.png)
Define the discrete-time dynamical system (1) as follows:
Jacobian matrix:
The non-trivial fixed point:
The linear approximation at X0(r):
The eigenvalues of the linear approximation A:
For 𝓇>5/4 the eigenvalues are complex, so the following shift is made in the growth rate 𝓇 to handle this restriction, and then the eigenvalues of A are computed again:
The value 𝓇0 where the fixed point X0(𝓇0) loses its stability, giving rise to a Neimark-Sacker bifurcation, is given by:
Therefore, the fixed point X0(𝓇0) loses its stability at 𝓇=2 as verified below:
The linear approximation A(𝓇) at the critical bifurcation value 𝓇=2 and its transpose:
The critical multipliers are given by:
The transversality condition:
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/75a1976c7c810f74.png)
The nondegeneracy condition:
ζn≠1, for n=1,2,3,4.
The critical eigenvectors q,
, p,
:
The normalization 〈pn,q〉=1
The bilinear and trilinear forms to compute the first Lyapunov coefficient:
The first Lyapunov coefficient (additional nondegeneracy condition) is given by
![](https://www.wolframcloud.com/obj/resourcesystem/images/d71/d71db709-5757-44e8-a522-45a3bdfaf3d6/1-0-1/34926cabe51e1f0d.png)
Finally, compute the first Lyapunov coefficient:
Therefore, a unique and stable closed invariant curve bifurcates from the non-trivial fixed point X0(𝓇) for 𝓇>2.
Quadratic coefficients of the normal form for Bogdanov-Takens bifurcation (15)
Compute the quadratic coefficients of the Bogdanov-Takens bifurcation normal form for a predator-prey model with non-linear predator reproduction and prey competition:
The Jacobian matrix:
Look for the Bogdanov-Takens point:
To explicitly handle constraint
, apply the following shift to get the Bogdanov-Takens point:
With the above shift in the predator population, the equilibrium point of coexistence and the critical value of bifurcation are as follows:
The linear approximation at x0(ζ0):
Verify that J0(ζ0) has a pair of null eigenvalues:
The critical right eigenvectors q0 and q1:
Verify the conditions 〈J0(ζ0),q0〉=0 y 〈J0(ζ0),q1〉=q0
The transition matrix Q with the critical right eigenvectors:
The critical left eigenvectors p0 and p1:
Verify the conditions 〈J0(ζ0)T,p1〉=0 y 〈J0(ζ0)T,p0〉=p1
Finally, verify the conditions 〈q0,p0〉=〈q1,p1〉=1 y 〈q0,p1〉=〈q1,p0〉=0 and that the matrix J0(ζ0) is similar to a simple Jordan block:
The quadratic coefficients, involved in the nondegeneracy condition, can be computed in the following two ways:
The Bogdanov-Takens bifurcation in this case is non-degenerate when ζ0!=2 and the unique point of degeneracy is: