Determining codimension of Bogdanov-Takens and Bautin bifurcations via simplest normal form computation

Abstract

In solving real-world problems, determining the codimension of Bogdanov-Takens (BT) and Bautin (generalized Hopf) bifurcations can be very challenging, even for simple two-dimensional dynamical systems. This difficulty becomes particularly evident when the number of system parameters exceeds the codimension of the bifurcations. Such challenges are closely linked to analyzing complex dynamics, such as the bifurcation of multiple limit cycles and homoclinic/heteroclinic bifurcations. In this paper, we use two population systems to demonstrate a systematic approach for determining the conditions that define the codimension of BT and Bautin bifurcations.

Figures (9)

• Figure 1: Codimension-3 BT bifurcation diagrams based on the normal form \ref{['Eqn25']}. (a)–(b) Bifurcation surfaces: red indicates saddle-node (SN) bifurcations, blue indicates Hopf bifurcations, and green indicates homoclinic loop (HL) bifurcations. The red curve on the blue surface marks the generalized Hopf (GH) bifurcation, while the blue curve on the green surface corresponds to the degenerate homoclinic loop (DHL) bifurcation. (c)–(d) Projections of the three-dimensional bifurcation diagrams onto the cone intersecting the $2$-sphere $\beta_1^2+\beta_2^2+\beta_3^2=\sigma^2$ with $\sigma=0.1$. In (c), curves are obtained from the bifurcation formulas in Theorem \ref{['Thm3']}: the intersection of the pink and blue curves indicates the GH bifurcation, while the intersection of the brown and green curves marks the DHL bifurcation. (d) A schematic representation of the bifurcation diagram with the corresponding phase portraits.
• Figure 2: Curve of $v_{1a} = 0$ in the $g$–$X_2$ plane, illustrating the continuum of parameter pairs $(g, X_2)$ that yield two small-amplitude limit cycles. Any point on the green segment between the blue line and the red curve corresponds to values of $g$ and $X_2$ for which this phenomenon occurs. A specal case, $(g,X_2)=(1,1+\frac{1}{2} \sqrt{2})$, is marked by the black circle.
• Figure 3: Bifurcation diagram of system \ref{['Eqn7']} projected onto the $e$–$X$ plane for $e=3\sqrt{2}-4$ and $g=1$. Here, SN, H, and GH denote the saddle-node, Hopf, and generalized Hopf bifurcations, respectively: (a) $n=\sqrt{2}-1$; (b) $n=2-\sqrt{2}$.
• Figure 4: Simulation of system \ref{['Eqn7']} for $g=1.004$ and $n=0.41071668$: (a) bifurcation diagram in the $e$–$X_2$ plane; the three vertical lines mark the values of $e$ used in the subsequent simulations. (b) $e=0.25223318$, trajectories converging to the stable equilibrium ${\rm E_2}$; (c) $e=0.24223719$, two limit cycles surrounding stable ${\rm E_2}$ (outer stable, inner unstable); (d) $e=0.24$, one stable limit cycle enclosing the unstable ${\rm E_2}$.
• Figure 5: BT bifurcation diagram of the original system \ref{['Eqn7']}, projected on the $g$-$e$ parameter plane, showing four points (a), (b), (c) and (d) for simulations taken along the line: $g=1.15$. The corresponding simulated phase portraits are shown in Figure \ref{['Fig6']}. SN, H, GH, HL and BT denote saddle-node, Hopf, generalized Hopf, homoclinic loop and Bogdanov-Takens bifurcations, respectively.

Theorems & Definitions (23)

• Remark 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Remark 2.4
• Theorem 2.5
• Theorem 2.6