In the theory of discrete dynamical systems, the loss of stability of a fixed point can occur in several ways. While the Fold bifurcation deals with an eigenvalue crossing $+1$ and the Flip bifurcation with an eigenvalue crossing $-1$ (period-doubling), the **Neimark-Sacker bifurcation** describes the scenario where a pair of complex conjugate eigenvalues crosses the unit circle. This is the discrete-time analogue of the [Hopf bifurcation](/pages/1047) in [continuous](/page/Continuity) systems: the unit circle $|z| = 1$ plays the role of the imaginary axis, and the invariant closed curve that emerges plays the role of the [limit](/page/Limit) cycle. The question the theory answers is: given a family of maps depending on a parameter $\alpha$, with a fixed point whose eigenvalues cross the unit circle as $\alpha$ crosses a critical value, does an invariant closed curve branch off from the fixed point? If so, is the curve attracting (nearby orbits spiral onto it) or repelling (nearby orbits are pushed away)? The answer depends on the interplay between the linear spectral crossing and the leading-order nonlinear terms, distilled into a single computable scalar — the First Lyapunov Coefficient — exactly as in the continuous case. ## Setup We consider a smooth family of maps on $\mathbb{R}^2$ with a real parameter $\alpha \in \mathbb{R}$: \begin{align*} x \mapsto f(x, \alpha), \quad x \in \mathbb{R}^2, \quad \alpha \in \mathbb{R}, \end{align*} where $f \in C^k(U \times V, \mathbb{R}^2)$ with $k \ge 5$, $U \subseteq \mathbb{R}^2$ a neighbourhood of the origin, and $V \subseteq \mathbb{R}$ a parameter interval. We assume that $x = 0$ is a fixed point for all parameter values: \begin{align*} f(0, \alpha) = 0 \quad \text{for every } \alpha \text{ near } 0. \end{align*} This is not restrictive: if a family of fixed points $x^*(\alpha)$ depends smoothly on $\alpha$, the coordinate change $y = x - x^*(\alpha)$ reduces to this form. The linearisation at the origin is the $\alpha$-dependent Jacobian matrix $A(\alpha) := D_x f(0, \alpha)$, whose eigenvalues govern the local stability of the fixed point. When the system is higher-dimensional ($x \in \mathbb{R}^n$ with $n > 2$), the [center manifold theorem](/page/Center%20Manifold%20Theorem) reduces the bifurcation analysis to a two-dimensional center manifold, just as in the continuous case. The essential dynamics are captured by the restriction of the map to this invariant surface. ## Spectral Hypothesis The bifurcation requires the eigenvalues of $A(\alpha)$ to cross the unit circle through a conjugate pair. [definition: Neimark-Sacker Spectral Hypothesis] The map $x \mapsto f(x, \alpha)$ satisfies the **Neimark-Sacker spectral hypothesis** at $\alpha = 0$ if the Jacobian $A(0) = D_x f(0, 0)$ has eigenvalues: \begin{align*} \lambda(0), \overline{\lambda(0)} = e^{\pm i\theta_0}, \quad \theta_0 \in (0, \pi), \end{align*} lying on the unit circle with non-zero imaginary part. [/definition] The condition $\theta_0 \in (0, \pi)$ excludes real eigenvalues: $\theta_0 = 0$ gives a double eigenvalue at $+1$ (Fold), and $\theta_0 = \pi$ gives a double eigenvalue at $-1$ (Flip). The eigenvalues $e^{\pm i\theta_0}$ generate a discrete rotation of $\mathbb{R}^2$ by angle $\theta_0$ per iterate, which is the linear mechanism behind the closed invariant curve. The key difference from the continuous Hopf setting is that the stability [boundary](/page/Boundary) is the unit circle $|\lambda| = 1$ rather than the imaginary axis $\mathrm{Re}(\lambda) = 0$. A fixed point of a map is stable when all eigenvalues satisfy $|\lambda| < 1$ (inside the unit disc) and unstable when any eigenvalue satisfies $|\lambda| > 1$. ## Complexification and Normal Form To analyse the nonlinear dynamics near the bifurcation, we pass to complex coordinates on $\mathbb{R}^2$. This requires two steps: first, a real change of basis that puts the linear part in rotation form; second, a complexification that diagonalises the rotation. ### Putting the Linear Part in Rotation Form The Jacobian $A(0)$ at the bifurcation has complex eigenvalues $\lambda, \bar{\lambda} = e^{\pm i\theta_0}$. A real $2 \times 2$ matrix with complex eigenvalues cannot be diagonalised over $\mathbb{R}$, but it can be brought to **rotation form** — the real [Jordan normal form](/theorems/864) for a conjugate pair. The procedure uses the eigenvectors. Since $A(0)$ is real but $\lambda$ is complex, the eigenvector $w$ satisfying $A(0)w = \lambda w$ is necessarily complex. Write it as $w = p + iq$ where $p, q \in \mathbb{R}^2$ are the real and imaginary parts. Define the change-of-basis matrix: \begin{align*} Q = \begin{pmatrix} | & | \\ p & q \\ | & | \end{pmatrix}. \end{align*} In the new coordinates $(u, v)^T = Q^{-1}(x, y)^T$, the linear part of the map becomes: \begin{align*} Q^{-1}A(0)Q = \begin{pmatrix} \cos\theta_0 & -\sin\theta_0 \\ \sin\theta_0 & \cos\theta_0 \end{pmatrix}, \end{align*} a rotation by angle $\theta_0$. This works because the eigenvector equation $A(0)(p + iq) = e^{i\theta_0}(p + iq)$ separates into $A(0)p = \cos\theta_0 \cdot p - \sin\theta_0 \cdot q$ and $A(0)q = \sin\theta_0 \cdot p + \cos\theta_0 \cdot q$, which is exactly the statement $Q^{-1}A(0)Q = R_{\theta_0}$. The nonlinear terms transform under the same coordinate change: if the original map is $\bar{x} = A(0)x + N(x)$ where $N$ collects the nonlinear terms, then in the new coordinates $(u,v)$: \begin{align*} \begin{pmatrix} \bar{u} \\ \bar{v} \end{pmatrix} = \begin{pmatrix} \cos\theta_0 & -\sin\theta_0 \\ \sin\theta_0 & \cos\theta_0 \end{pmatrix}\begin{pmatrix} u \\ v \end{pmatrix} + Q^{-1}N(Q\begin{pmatrix} u \\ v \end{pmatrix}). \end{align*} ### Complexification With the linear part in rotation form, define $z := u + iv$ and $\bar{z} = u - iv$, so that $u = (z + \bar{z})/2$ and $v = (z - \bar{z})/(2i)$. The rotation matrix acts on $z$ as multiplication by $e^{i\theta_0}$: \begin{align*} \bar{z} = \bar{u} + i\bar{v} = (\cos\theta_0 \cdot u - \sin\theta_0 \cdot v) + i(\sin\theta_0 \cdot u + \cos\theta_0 \cdot v) = e^{i\theta_0}(u + iv) = e^{i\theta_0}z. \end{align*} Including the nonlinear terms, the map takes the complex form: \begin{align*} \bar{z} = e^{i\theta_0} z + \sum_{j+k \ge 2} \frac{1}{j!\,k!}\,g_{jk}\,z^j\bar{z}^k, \end{align*} where the coefficients $g_{jk} \in \mathbb{C}$ are determined by re-expressing the transformed nonlinearity $Q^{-1}N(Q(\cdot))$ in terms of $z$ and $\bar{z}$. [example: Complexification of a Planar Map] Consider the map from Q5 of the 2023/2025 exam: $\bar{x} = y + x^3$, $\bar{y} = -x + by$ (at $a = 1$). The Jacobian is $A = \begin{pmatrix} 0 & 1 \\ -1 & b \end{pmatrix}$ with eigenvalue $\lambda = \frac{b}{2} - i\frac{\sqrt{4-b^2}}{2} = e^{-i\omega}$. **Step 1: Eigenvector.** Solve $(A - \lambda I)w = 0$. From the first row: $-\lambda w_1 + w_2 = 0$, so $w_2 = \lambda w_1$. Taking $w_1 = 1$: \begin{align*} w = \begin{pmatrix} 1 \\ \lambda \end{pmatrix} = \begin{pmatrix} 1 \\ b/2 \end{pmatrix} + i\begin{pmatrix} 0 \\ -\sqrt{4-b^2}/2 \end{pmatrix}. \end{align*} **Step 2: Change of basis.** $Q = \begin{pmatrix} 1 & 0 \\ b/2 & -\sqrt{4-b^2}/2 \end{pmatrix}$. In the new coordinates $(u,v)^T = Q^{-1}(x,y)^T$, the linear part becomes a rotation by $\omega$. Since the first column of $Q$ is $(1, b/2)^T$, the inverse gives $x = u$ — the original $x$-coordinate equals the new $u$-coordinate. **Step 3: Nonlinear terms.** The only nonlinearity is $x^3$, which in the new coordinates is $u^3$. Since $u = (z + \bar{z})/2$: \begin{align*} \bar{z} = e^{i\omega}z + \left(\frac{z + \bar{z}}{2}\right)^3. \end{align*} [/example] ## The First Lyapunov Coefficient At the critical parameter $\alpha = 0$ where $r(0) = 1$, the linear part of the map is an exact rotation and provides no information about the radial dynamics. In the continuous Hopf setting, a near-identity coordinate change eliminates all non-resonant terms from $\dot{z} = i\omega_0 z + \ldots$, leaving the normal form $\dot{z} = i\omega_0 z + c_1 z^2\bar{z} + O(|z|^4)$ where the resonant term $z^2\bar{z}$ (with divisor zero) is the only survivor. The coefficient $\operatorname{Re}(c_1) = L_1$ then determines stability via a straightforward polar computation: $\dot{r} = L_1 r^3 + O(r^5)$. For maps, the same procedure applies — eliminate non-resonant monomials by dividing by their divisors $e^{i(j-k)\theta_0} - e^{i\theta_0}$ — but requires the non-resonance condition to ensure all divisors are nonzero. The following theorem makes this precise and shows that the resulting normal form has the same structure as in the continuous case: a single resonant term $z|z|^2$ whose real coefficient $L_1$ controls the amplitude dynamics. [quotetheorem:927] The normal form separates the amplitude and phase dynamics cleanly: $L_1$ controls whether the radius grows or shrinks, while $\Omega_1$ is a nonlinear correction to the rotation frequency. This separation is why $L_1$ alone determines stability — the $\Omega_1$ term affects only how fast orbits wind around the origin, not whether they spiral inward or outward. The leading nonlinear correction to the amplitude is captured by a single real number, computed from the Taylor coefficients of the original map. [definition: First Lyapunov Coefficient (Discrete)] Consider the complex normal form of the map at $\alpha = 0$: \begin{align*} z \mapsto e^{i\theta_0} z + \tfrac{1}{2}g_{20}\,z^2 + g_{11}\,z\bar{z} + \tfrac{1}{2}g_{02}\,\bar{z}^2 + \tfrac{1}{2}g_{21}\,z^2\bar{z} + \cdots \end{align*} The **First Lyapunov Coefficient** $L_1 \in \mathbb{R}$ is: \begin{align*} L_1 := \mathrm{Re}\!\left(\frac{e^{-i\theta_0} g_{21}}{2}\right) - \mathrm{Re}\!\left(\frac{(1 - 2e^{i\theta_0})\,e^{-2i\theta_0}}{2(1 - e^{i\theta_0})}\,g_{20}\,g_{11}\right) - \frac{1}{2}|g_{11}|^2 - \frac{1}{4}|g_{02}|^2. \end{align*} [/definition] The formula is more involved than its continuous counterpart because eliminating the quadratic terms from a map requires solving a different cohomological equation. In the continuous case, the near-identity transformation removes monomials $z^j\bar{z}^k$ whose associated "divisor" $i(j-k-1)\omega_0$ is non-zero. In the discrete case, the divisor is $e^{i(j-k)\theta_0} - e^{i\theta_0}$, and the non-resonance condition $e^{ik\theta_0} \neq 1$ for $k = 1,2,3,4$ ensures these divisors do not vanish for the relevant quadratic and cubic monomials. The four terms in the $L_1$ formula reflect the direct cubic contribution ($g_{21}$ term) and the three second-order interaction terms generated by eliminating the quadratic monomials $z^2$, $z\bar{z}$, and $\bar{z}^2$. ## The Neimark-Sacker Bifurcation Theorem [definition: Transversality Condition (Discrete)] The map satisfies the **transversality condition** at $\alpha = 0$ if the modulus of the crossing eigenvalue moves through the unit circle at non-zero speed: \begin{align*} r'(0) := \frac{d}{d\alpha}|\lambda(\alpha)|\bigg|_{\alpha=0} \neq 0. \end{align*} [/definition] The transversality condition ensures that $\alpha = 0$ is a genuine crossing of the unit circle, not a tangency. Combined with the spectral hypothesis, non-resonance, and non-degeneracy of $L_1$, it yields the full bifurcation result. In the continuous [Hopf bifurcation](/page/Hopf%20Bifurcation), the First Lyapunov Coefficient $L_1$ emerges from studying how the amplitude $r$ changes over one full rotation: the linear part $\dot{z} = i\omega_0 z$ generates continuous rotation, and integrating $\dot{r}$ over $[0, 2\pi]$ kills all non-resonant terms, leaving only the $z^2\bar{z}$ contribution. For maps, one iterate rotates by a fixed angle $\theta_0$ rather than sweeping continuously, so a single-step [integral](/page/Integral) is not available. Instead, the same averaging occurs over many iterates: when $\theta_0/(2\pi)$ is irrational, Weyl's equidistribution theorem guarantees the angles $\theta_0, 2\theta_0, 3\theta_0, \ldots$ fill the circle uniformly, reproducing the continuous integral in the long-time average. The non-resonance condition (NS3) ensures precisely this — that the discrete rotation visits the entire circle uniformly rather than sitting on a finite (measure-zero) subset, which would allow extra monomials beyond $z^2\bar{z}$ to survive the averaging. When it holds, $L_1$ controls the amplitude dynamics just as in the continuous case, and the following theorem gives the complete local picture. [quotetheorem:235] The supercritical case ($L_1 < 0$) is a "soft" transition: as $\alpha$ increases past zero, the fixed point smoothly gives way to a small closed curve on which orbits execute quasi-periodic (or phase-locked) motion. The subcritical case ($L_1 > 0$) is "hard": the fixed point loses stability abruptly, and orbits can jump to a distant attractor. ## Resonances The non-resonance condition (NS3) excludes $e^{ik\theta_0} = 1$ for $k \le 4$. When it fails, the normal form transformation breaks down because the cohomological equation has a zero divisor. The consequences depend on the order of resonance. [remark: Why Rational Rotation Angles Cause Problems] The non-resonance condition has a concrete dynamical origin. For a continuous system (ODE), the flow rotates continuously through all angles, and the net effect of a nonlinear term on the amplitude is obtained by integrating over one full rotation: $\int_0^{2\pi} e^{in\theta}\, d\theta = 0$ for $n \neq 0$. Every non-resonant term averages to zero, and no conditions on $\omega_0$ are needed. For a map, the rotation is **discrete**: each iterate rotates by $\theta_0$, so the orbit visits angles $\theta_0, 2\theta_0, 3\theta_0, \ldots$ The net effect on the amplitude is determined by averaging over these discrete angles. When $\theta_0/(2\pi)$ is irrational, Weyl's equidistribution theorem guarantees these angles fill $[0, 2\pi)$ uniformly, and the average reproduces the integral — non-resonant terms average to zero, just as in the continuous case. When $\theta_0 = 2\pi p/q$ for a small integer $q$, the orbit visits only $q$ equally spaced angles. The average becomes a finite sum $\frac{1}{q}\sum_{n=0}^{q-1} e^{in \cdot m \cdot 2\pi/q}$, which is nonzero whenever $m$ is a multiple of $q$. This means additional monomials — beyond $z^2\bar{z}$ — produce a systematic amplitude change that does not cancel. For $q = 3$, the extra resonant monomial $\bar{z}^2$ is quadratic, dominating the cubic $z^2\bar{z}$. For $q = 4$, the extra resonant monomial $\bar{z}^3$ is cubic, competing with $z^2\bar{z}$ at the same order. In both cases, $L_1$ alone does not capture the leading behaviour. Algebraically, this manifests as a zero divisor in the normal form transformation: to eliminate a monomial $z^j\bar{z}^k$, one must divide by $e^{i(j-k)\theta_0} - e^{i\theta_0}$, which vanishes precisely when the corresponding average is nonzero. The non-resonance condition $e^{ik\theta_0} \neq 1$ for $k \le 4$ ensures that all quadratic and cubic divisors are nonzero, so that $z^2\bar{z}$ is genuinely the leading resonant term — just as it always is in the continuous case. [/remark] [remark: Strong Resonances ($k \le 4$)] If the non-resonance condition is violated, the Neimark-Sacker theorem as stated does not apply: - **$k=1$ ($\theta_0 = 0$, eigenvalue $+1$).** This is a Fold or Transcritical bifurcation of the fixed point — the eigenvalues are real and the mechanism is fundamentally different. - **$k=2$ ($\theta_0 = \pi$, eigenvalue $-1$).** This is a Flip (period-doubling) bifurcation: a period-2 orbit branches from the fixed point, not an invariant curve. - **$k=3$ ($\theta_0 = 2\pi/3$, 1:3 resonance).** The normal form contains a resonant quadratic term $\bar{z}^2$ that cannot be eliminated. Generically, no invariant circle emerges; instead, a complicated local bifurcation structure appears, typically involving saddle-node bifurcations of period-3 orbits. - **$k=4$ ($\theta_0 = \pi/2$, 1:4 resonance).** The normal form contains a resonant cubic term $\bar{z}^3$. The dynamics are richer: invariant curves may or may not exist depending on higher-order coefficients, and the local phase portrait can include period-4 orbits of various stability types. Strong resonances are codimension-two phenomena when the resonance angle $\theta_0$ is treated as an additional parameter alongside $\alpha$. [/remark] [remark: Why Exactly $k \le 4$] The cutoff at $k = 4$ is not arbitrary — it is determined by the order at which $L_1$ operates. Computing $L_1$ requires eliminating all **quadratic** monomials ($z^2, z\bar{z}, \bar{z}^2$) and all **non-resonant cubic** monomials ($z^3, z\bar{z}^2, \bar{z}^3$) from the normal form. Each elimination divides by a divisor $e^{i(j-k)\theta_0} - e^{i\theta_0}$, which vanishes when $e^{i(j-k-1)\theta_0} = 1$. The values of $|j - k - 1|$ that appear across all six monomials are exactly $1, 2, 3, 4$: | Monomial | $(j,k)$ | $j - k - 1$ | Divisor zero when | |---|---|---|---| | $z^2$ | $(2,0)$ | $1$ | $e^{i\theta_0} = 1$ | | $z\bar{z}$ | $(1,1)$ | $-1$ | $e^{-i\theta_0} = 1$ | | $\bar{z}^2$ | $(0,2)$ | $-3$ | $e^{-3i\theta_0} = 1$ | | $z^3$ | $(3,0)$ | $2$ | $e^{2i\theta_0} = 1$ | | $z\bar{z}^2$ | $(1,2)$ | $-2$ | $e^{-2i\theta_0} = 1$ | | $\bar{z}^3$ | $(0,3)$ | $-4$ | $e^{-4i\theta_0} = 1$ | For $k \ge 5$ (e.g., $\theta_0 = 2\pi/5$), the extra resonances affect monomials of order $\ge 4$, which are beyond the reach of the $L_1$ computation. The invariant curve still exists and its stability is determined by $L_1$; the higher-order resonances produce only weak effects (phase-locking on the curve, Arnold tongues) without invalidating the leading-order analysis. [/remark] ### Dynamics on the invariant curve When the invariant curve $\Gamma_\alpha$ exists (the non-resonant or weakly resonant case), the dynamics of the map restricted to $\Gamma_\alpha$ are a circle map. The rotation number $\rho$ of this circle map determines the qualitative behaviour. [remark: Weak Resonances and Phase Locking] Suppose $\theta_0/(2\pi)$ is irrational. Then, for $\alpha$ sufficiently small, the restriction $f|_{\Gamma_\alpha}$ is topologically conjugate to an irrational rotation, and every orbit on $\Gamma_\alpha$ is dense — this is **quasi-periodic motion**. If $\theta_0 = 2\pi p/q$ with $\gcd(p,q) = 1$ and $q \ge 5$ (a **weak resonance**), the invariant curve persists but the dynamics on it are qualitatively different. By the Poincaré-Birkhoff theorem, the map restricted to $\Gamma_\alpha$ generically has a finite even number of period-$q$ orbits: half are stable (nodes on the curve) and half are unstable (saddles on the curve). Orbits are attracted to the stable period-$q$ points rather than filling the curve densely — this is **phase locking**. In parameter space, the regions where the rotation number is locked to a rational value $p/q$ form wedge-shaped domains called **Arnold tongues**, emanating from the point $(\alpha, \theta) = (0, 2\pi p/q)$. Between Arnold tongues, the rotation number is irrational and the motion is quasi-periodic. This intricate alternation of locked and quasi-periodic regimes is a hallmark of the Neimark-Sacker bifurcation that has no analogue in the continuous Hopf bifurcation. [/remark] ## Comparison With the Hopf Bifurcation The Neimark-Sacker bifurcation is the discrete analogue of the [Hopf bifurcation](/pages/1047), and the parallels are extensive: | Feature | Hopf (continuous) | Neimark-Sacker (discrete) | |---|---|---| | System type | ODE: $\dot{X} = F(X, \mu)$ | Map: $x \mapsto f(x, \alpha)$ | | Stability boundary | Imaginary axis $\mathrm{Re}(\lambda) = 0$ | Unit circle $|\lambda| = 1$ | | Crossing eigenvalues | $\pm i\omega_0$ | $e^{\pm i\theta_0}$ | | Bifurcating object | Limit cycle (periodic orbit) | Invariant closed curve | | Dynamics on object | Periodic (single frequency) | Quasi-periodic or phase-locked | | Key scalar | $L_1 = \frac{1}{2\omega_0}\mathrm{Re}(ig_{20}g_{11} + \omega_0 g_{21})$ | $L_1$ (four-term formula above) | | Resonance issue | None (continuous rotation) | Strong ($k \le 4$) and weak ($k \ge 5$) | The most significant difference is the resonance structure. In the continuous case, the eigenvalue $e^{i\omega_0 t}$ generates a flow that never returns to rational rotations (since $\omega_0 t$ sweeps continuously), so resonances do not arise. In the discrete case, the eigenvalue $e^{i\theta_0}$ generates a discrete rotation that can be exactly periodic when $\theta_0/(2\pi)$ is rational, leading to the Arnold tongue structure described above. ## Examples [example: Supercritical Neimark-Sacker Bifurcation] Consider the planar map in complex coordinates: \begin{align*} z \mapsto (1 + \alpha)\,e^{i\theta_0}\,z - z^2\bar{z}, \end{align*} with $\theta_0 \notin \{0, \pi/2, 2\pi/3, \pi\}$. **Step 1: Fixed point and eigenvalues.** The origin $z = 0$ is a fixed point for all $\alpha$, with multiplier $\lambda(\alpha) = (1+\alpha)e^{i\theta_0}$. The modulus is $r(\alpha) = 1 + \alpha$, so $r(0) = 1$ (criticality) and $r'(0) = 1 \neq 0$ (transversality). **Step 2: Taylor coefficients.** The map is already in normal form. The quadratic terms vanish: $g_{20} = g_{11} = g_{02} = 0$. The cubic coefficient satisfies $\frac{1}{2}g_{21} = -1$, so $g_{21} = -2$. **Step 3: Lyapunov coefficient.** Since $g_{20} = g_{11} = g_{02} = 0$: \begin{align*} L_1 = \mathrm{Re}\!\left(\frac{e^{-i\theta_0}(-2)}{2}\right) - 0 - 0 - 0 = \mathrm{Re}(-e^{-i\theta_0}) = -\cos\theta_0. \end{align*} For $\theta_0 \in (0, \pi/2)$, we have $\cos\theta_0 > 0$, so $L_1 < 0$: supercritical. For $\theta_0 \in (\pi/2, \pi)$, $L_1 > 0$: subcritical. **Step 4: Verification.** Writing $z_n = r_n e^{i\varphi_n}$, the modulus satisfies: \begin{align*} r_{n+1} = |z_{n+1}| = |(1+\alpha)e^{i\theta_0} z_n - z_n^2\bar{z}_n| = r_n|(1+\alpha)e^{i\theta_0} - r_n^2 e^{-i\varphi_n + 2i\varphi_n}|. \end{align*} For the simplified case where $z_n^2\bar{z}_n = r_n^2 z_n$ (which holds since $z^2\bar{z} = |z|^2 z$): \begin{align*} z_{n+1} = ((1+\alpha)e^{i\theta_0} - r_n^2)\,z_n, \end{align*} giving $r_{n+1} = r_n |1 + \alpha - r_n^2|$. An invariant curve has $r_{n+1} = r_n$, requiring $|1 + \alpha - r^2| = 1$. For $\alpha > 0$ small and $r^2 = \alpha$, we get $|1| = 1$ — so $r = \sqrt{\alpha}$ is an invariant circle. Since $r'(0) = 1 > 0$ and $L_1 = -\cos\theta_0 < 0$ (for $\theta_0 \in (0,\pi/2)$), this matches the supercritical prediction. [/example] [example: Hénon-Like Map Near Neimark-Sacker] Consider the two-dimensional map: \begin{align*} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} \alpha + x_2 - x_1^2 \\ \beta x_1 \end{pmatrix}, \end{align*} with parameters $(\alpha, \beta)$. The fixed points satisfy $x_1 = \alpha + x_2 - x_1^2$ and $x_2 = \beta x_1$, giving $x_1 = \alpha + \beta x_1 - x_1^2$ or $x_1^2 - (\beta - 1)x_1 - \alpha = 0$. The Jacobian at a fixed point $(x_1^*, \beta x_1^*)$ is: \begin{align*} A = \begin{pmatrix} -2x_1^* & 1 \\ \beta & 0 \end{pmatrix}, \end{align*} with eigenvalues satisfying $\lambda^2 + 2x_1^*\lambda - \beta = 0$, giving $\lambda = -x_1^* \pm \sqrt{(x_1^*)^2 + \beta}$. When $(x_1^*)^2 + \beta < 0$ (requiring $\beta < 0$), the eigenvalues are complex conjugate with modulus $|\lambda|^2 = -\beta = |\beta|$. The Neimark-Sacker condition $|\lambda| = 1$ requires $\beta = -1$, and the argument is $\theta_0 = \arccos(-x_1^*)$. This shows that the Neimark-Sacker bifurcation occurs along the curve $\beta = -1$ in the $(\alpha, \beta)$-parameter plane, with the angle $\theta_0$ varying as $\alpha$ changes (through its effect on $x_1^*$). The transition from a stable fixed point to quasi-periodic motion on an invariant curve is a standard route to complex dynamics in such maps. [/example] [example: Discrete Predator-Prey Model] Consider a discretisation of the Lotka-Volterra system: \begin{align*} x_{n+1} &= x_n \exp(r(1 - x_n) - a\,y_n), \\ y_{n+1} &= x_n(1 - \exp(-a\,y_n)), \end{align*} modelling prey $x$ and predator $y$ with growth rate $r$ and interaction parameter $a$. For appropriate parameter values, the coexistence fixed point $(x^*, y^*)$ has complex conjugate eigenvalues. As $r$ increases past a critical value $r^*$, the eigenvalue modulus crosses $1$ and a Neimark-Sacker bifurcation occurs: the stable equilibrium gives way to an invariant closed curve on which the populations oscillate quasi-periodically. This is a discrete analogue of the predator-prey limit cycles predicted by the continuous Hopf bifurcation, but with the added possibility of phase locking at rational rotation numbers. [/example] ## Problems [problem] Consider the complex map: \begin{align*} z \mapsto e^{i\pi/3}(1 + \alpha)\,z + c\,z^2\bar{z}, \end{align*} where $c \in \mathbb{C}$ is a constant and $\alpha \in \mathbb{R}$ is a parameter. 1. Verify that the non-resonance condition (NS3) fails. Identify the resonance order $k$. 2. Explain why the Neimark-Sacker theorem does not apply and what type of dynamics may be expected instead. [/problem] [solution] **Part 1.** The angle is $\theta_0 = \pi/3$. We check: $e^{ik\theta_0} = e^{ik\pi/3}$. For $k = 3$: $e^{3i\pi/3} = e^{i\pi} = -1 \neq 1$. For $k = 6$: $e^{6i\pi/3} = e^{2i\pi} = 1$. But we only need $k \in \{1,2,3,4\}$: $e^{i\pi/3} \neq 1$, $e^{2i\pi/3} \neq 1$, $e^{i\pi} = -1 \neq 1$, $e^{4i\pi/3} \neq 1$. So the non-resonance condition is actually satisfied for $k \le 4$, and the Neimark-Sacker theorem does apply. Wait — let us recheck. $\theta_0 = \pi/3$ corresponds to a $1{:}6$ rotation ($e^{6i\pi/3} = 1$), which is a weak resonance with $q = 6 \ge 5$. The strong resonance condition only excludes $k \le 4$. Since $e^{ik\pi/3} \neq 1$ for $k = 1,2,3,4$, the non-resonance condition (NS3) **is satisfied**, and the theorem applies. **Part 2.** Since (NS3) holds, the Neimark-Sacker theorem does apply. However, because $\theta_0/(2\pi) = 1/6$ is rational, the invariant curve $\Gamma_\alpha$ will generically exhibit phase locking: the map restricted to $\Gamma_\alpha$ will have period-6 orbits (half stable, half unstable) rather than dense quasi-periodic motion. The Arnold tongue for the $1{:}6$ resonance has non-zero width in parameter space. [/solution] [problem] Consider the complex map: \begin{align*} z \mapsto (1 + \alpha)\,e^{i\theta_0}\,z + g_{20}\,z^2 + g_{02}\,\bar{z}^2 + g_{21}\,z^2\bar{z}, \end{align*} with $\theta_0 = 1$ (irrational multiple of $2\pi$), $g_{20} = i$, $g_{02} = 1$, $g_{11} = 0$, and $g_{21} = -3e^{i}$. 1. Verify the non-resonance condition. 2. Compute the First Lyapunov Coefficient $L_1$. 3. Determine whether the bifurcation is supercritical or subcritical, and describe the resulting dynamics. [/problem] [solution] **Part 1.** Since $\theta_0 = 1$ radian and $1/(2\pi) \approx 0.1592$ is irrational, $e^{ik\theta_0} = e^{ik} \neq 1$ for any integer $k$ (in particular for $k = 1,2,3,4$). The non-resonance condition is satisfied. **Part 2.** With $g_{20} = i$, $g_{11} = 0$, $g_{02} = 1$, $g_{21} = -3e^{i}$: \begin{align*} L_1 &= \mathrm{Re}\!\left(\frac{e^{-i}(-3e^{i})}{2}\right) - \mathrm{Re}\!\left(\frac{(1 - 2e^{i})e^{-2i}}{2(1 - e^{i})}\cdot i \cdot 0\right) - \frac{1}{2}|0|^2 - \frac{1}{4}|1|^2 \\ &= \mathrm{Re}\!\left(\frac{-3}{2}\right) - 0 - 0 - \frac{1}{4} \\ &= -\frac{3}{2} - \frac{1}{4} = -\frac{7}{4}. \end{align*} **Part 3.** Since $L_1 = -7/4 < 0$, the bifurcation is **supercritical**: a stable invariant closed curve emerges for $\alpha > 0$. Because $\theta_0/(2\pi)$ is irrational, the dynamics on the invariant curve are quasi-periodic — every orbit is dense on $\Gamma_\alpha$. The fixed point is stable for $\alpha < 0$ and unstable for $\alpha > 0$, with the attracting invariant curve replacing it as the asymptotic attractor. [/solution] ## References 1. Y. A. Kuznetsov, *Elements of Applied Bifurcation Theory*, 3rd ed., Springer (2004). §4.7. 2. J. Guckenheimer and P. Holmes, *Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields*, Springer (1983). §3.5. 3. S. Wiggins, *Introduction to Applied Nonlinear Dynamical Systems and Chaos*, 2nd ed., Springer (2003). §20. 4. V. I. Arnold, *Geometrical Methods in the Theory of Ordinary Differential Equations*, 2nd ed., Springer (1988). Ch. 5. 5. D. G. Aronson, M. A. Chory, G. R. Hall, and R. P. McGehee, "Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study," *Comm. Math. Phys.* **83** (1982), 303–354.