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Near two homoclinic orbits of a planar saddle equilibrium with contracting ratio $\delta >1$, the signed Poincaré return map has separate offsets $\mu ,\nu$. Two single-lobe periodic orbits exist for $\mu <0$ and $\nu <0$, respectively. A two-lobe periodic orbit is represented by positive $p,q$ solving the displayed equations. Its boundary consists of the compound homoclinic orbit curves $\mu =-A{\nu }^{\delta }$ for $\nu >0$ and $\nu =-B{\mu }^{\delta }$ for $\mu >0$, with $A,B>0$. In a sufficiently small return domain the derivatives tend to zero, so each admissible symbolic branch has at most one attracting periodic orbit. Restricting to $\mu =\nu$, $A=B$ gives the usual symmetric homoclinic gluing bifurcation.

An averaged amplitude equation describes the slow evolution of a periodic orbit's amplitude under a weak perturbation. If unperturbed energy is $H(r)$ with $H^{\prime }(r)\ne 0$, divide the mean energy drift over one fast period by $H^{\prime }(r)$. Simple zeros identify attracting or repelling persistent periodic orbits according to the derivative of this drift; this approximation is local to an appropriate perturbative amplitude range.

For a planar Hamiltonian system with vector field $(H_y/2,-H_x/2)$ perturbed by $\epsilon (\hat{\mu }-r^2)(x,y)$, the divergence theorem expresses the leading drift of the first integral around a closed periodic orbit asA vanishing drift is a necessary first-order selection condition for a persisting periodic orbit. A simple zero with outward drift on its inner side and inward drift on its outer side yields an attracting limit cycle. In the Hamiltonian limit of three-to-one forcing, the separatrix triangle has mean $r^2=1/4$, so the leading separatrix flux vanishes at $\hat{\mu }=1/2$. A homoclinic or heteroclinic transition still requires the separatrix splitting and higher-order corrections to be controlled.

Chaos describes sustained irregular dynamics with sensitive dependence in a deterministic system. The precise convention matters. Devaney chaos requires topological transitivity, dense periodic orbits and sensitive dependence on an invariant set. A common operational flow description asks for bounded nonperiodic long-term motion with sensitive dependence. These formulations are useful in different settings and are not universally equivalent. A positive largest Lyapunov exponent is a numerical diagnostic of exponential separation, not by itself a proof of every topological definition.

The displayed map has a symmetric periodic orbit $\{\pm \sqrt{\mu +1}\}$ for $\mu >-1$, with multiplier $(-2\mu -3{)}^2>1$. For $\mu >2$, put $s_{\pm }=(\mu \pm \sqrt{{\mu }^2-4})/2$. The two additional period-two orbits are $\{\sqrt{s_{+}},\sqrt{s_{-}}\}$ and their negatives. Their multiplier is $9-2{\mu }^2$, so they are stable for $2<\mu <\sqrt{5}$ and undergo a period-doubling bifurcation at $\sqrt{5}$. At $\mu =2$ these points coincide with fixed points, not genuine two-cycles.

A multiplier $B(x,y)$ whose rescaled planar vector field has divergence of one strict sign can exclude periodic orbits in a simply connected region. For a system $\dot{x}=xF$, $\dot{y}=yG$ in the positive quadrant, $B=1/(xy)$ is often effective.

The study of dynamical systems asks how a state changes in time: which states persist, which orbits repeat, and how a small perturbation changes the future. A dynamical system combines a state space with an evolution rule. For a seasonal population, the state might be its abundance and one step one generation; for a harmonic oscillator, a state vector includes both position and velocity. Choosing a sufficient state is essential: position alone does not determine where an oscillator moves next.

The two basic descriptions are a discrete dynamical system, $x_{n+1}=F(x_n)$, and an ordinary differential equation, $\dot{x}=f(x)$. Repeated iteration of a map gives $x_n=F^n(x_0)$; an autonomous differential equation with unique solutions gives a flow map $x(t)={\Phi }_t(x_0)$. In either case, evolving for two successive durations agrees with evolving for their sum. The logistic map provides a particularly simple discrete dynamical system; the Lorenz system supplies a continuous-time example with a very different geometry.

Deterministic evolution says that the exact rule and exact initial condition specify one future. Predicting that future from measurements is a separate question. An instrument supplies a range of possible initial conditions, and a computation adds small errors. Near an asymptotically stable fixed point, nearby orbits converge and the long-term state can become easier to predict. With sensitive dependence on initial conditions, arbitrarily close starting states can eventually lead to a visible disagreement, even though each follows the same rule.

Much of the usefulness of dynamical systems comes from describing outcomes without writing every solution explicitly. Fixed points identify possible resting states; periodic orbits identify repeating behaviour; stability theory asks whether those behaviours survive small disturbances. A bifurcation diagram shows how changing a bifurcation parameter changes the possible outcomes. This is useful in population dynamics, where a stable abundance and an alternating abundance have different consequences, and in thermal convection, where a steady flow and a changing flow transport heat differently.

The beauty of the subject is that a small formula can organize a large variety of behaviour. The logistic map moves from an attracting fixed point to attracting periodic orbits and then to parameter values supporting sensitive motion. Its period-doubling cascade has a recognizable structure across many unimodal maps. Nevertheless, Devaney chaos, Glendinning chaos, and a positive Lyapunov exponent express different properties; a claim of chaos should specify its definition, its state space, and the parameter values under consideration.

For a weak perturbation of a planar Hamiltonian system, integrate the first-order change of the unperturbed Hamiltonian around each unperturbed periodic orbit. Zeros of this averaged energy drift select candidate perturbed periodic orbits, and a change from positive to negative drift indicates stability.

For $\dot{x}=y+ax-b{x}^3$, $\dot{y}=x^3-x$, the function $H=y^2/2+x^2/2-x^4/4$ obeys $\dot{H}=(x-x^3)(ax-b{x}^3)$. Every nonconstant periodic orbit lies in $|x|<1$: at a maximum above one or a minimum below minus one, $\ddot{x}=x^3-x$ has the wrong sign; equality would force the solution to be an equilibrium by uniqueness. If $a\ne 0$ and $b/a<1$, then $\dot{H}=a{x}^2(1-x^2)(1-(b/a)x^2)$ has the strict sign of $a$ except at $x=0$, preventing a nonconstant periodic orbit.

For positive $\sigma ,b,r$, the origin of the Lorenz system loses stability at $r=1$ and two equilibria $(\pm \sqrt{b(r-1)},\pm \sqrt{b(r-1)},r-1)$ emerge stably. Center-manifold projection gives $\dot{q}=\sigma (r-1)q/(\sigma +1)-\sigma q^3/[b(\sigma +1)]+\cdots$, proving a supercritical pitchfork. The nonzero-branch characteristic polynomial is ${\lambda }^3+(\sigma +b+1){\lambda }^2+b(\sigma +r)\lambda +2\sigma b(r-1)$. If $\sigma \le b+1$ these equilibria remain stable for all $r>1$. Otherwise their transverse Hopf crossing occurs at the displayed parameter, with frequency squared $b(\sigma +r_H)$. Its cubic Hopf coefficient is positive, so this Hopf is subcritical, producing an unstable small periodic orbit on the stable-equilibrium side. A zero eigenvalue detects the pitchfork, while the Hopf condition uses the Routh-Hurwitz equality; neither alone determines nonlinear criticality without the corresponding normal-form calculation.

For $\dot{x}=2x-x^2-y$, $\dot{y}=y-kx$ with $k>2$, take the positive Dulac function in the title. More generally, $\psi =e^{a(x+y)}$ givesThe equal coefficients in the exponential cancel the $y$ term. Setting $a=-2/(k-2)$ also cancels the linear $x$ term, leaving $\psi (3+2x^2/(k-2))>0$. The Bendixson-Dulac theorem excludes periodic orbits in the open positive quadrant. The axes are not invariant: the vector field points out of the quadrant at every nonzero axis point. A closed orbit staying in the nonnegative quadrant therefore cannot touch an axis, and cannot pass through the equilibrium at the origin by uniqueness. This extends the exclusion to the closed quadrant. Choosing an exponential multiplier turns a differential search for a useful Dulac function into cancellation of polynomial coefficients.

On the punctured plane, the vector field $\dot{x}=(1/r-1)x-y$, $\dot{y}=(1/r-1)y+x$ has the displayed polar equations. Its exact trajectories satisfy $r(t)=1+(r(0)-1)e^{-t}$ and $\theta (t)=\theta (0)+t$. Every forward trajectory approaches the unit circle, rotating counterclockwise, from inside or outside according to its initial radius. The circle is an attracting limit cycle with least period $2\pi$. The origin is not an equilibrium: the original vector field is undefined there. The singularity prevents an application of a whole-plane Bendixson-Dulac criterion to this periodic orbit.

The amplitude equations $u^{\prime }=-2uv$, $v^{\prime }={\lambda }_2+v^2+u^2$ satisfy $u^{\prime }=F_v$, $v^{\prime }=-F_u$, hence preserve the displayed first integral. For ${\lambda }_2=-a^2<0$ and $u\ge 0$, the centre equilibrium is $(a,0)$ with $F=2a^3/3$, while the two boundary saddle equilibria are $(0,\pm a)$ with $F=0$. The level $F=0$ contains the boundary connection and the arc $u^2=3(a^2-v^2)$ joining the two saddle equilibria. Levels $0<F<2a^3/3$ form closed periodic orbits. Adding an independent rotating phase turns these closed amplitude trajectories into invariant tori.

The first Lyapunov coefficient is the cubic radial coefficient in the center-manifold normal form of a smooth Hopf bifurcation, after an appropriate normalization. In $\dot{r}=\beta r+l_1{r}^3+\cdots$, its nonzero sign distinguishes stable and unstable bifurcating small cycles relative to the eigenvalue-crossing direction. If $l_1=0$, higher-order terms are required and the generic cubic Hopf theorem does not apply.

For a Floquet multiplier $m\ne 0$ of a period-$T$ linear variational equation, a Floquet exponent is any logarithmic rate $\rho$ satisfying the displayed relation. Its real part is uniquely $T^{-1}\log |m|$, while its imaginary part is defined modulo $2\pi /T$. Negative transverse real parts give local attraction to a periodic orbit; the autonomous phase direction has multiplier one and exponent zero. Constant-coefficient amplitude equations supply exponents directly as the eigenvalues of their Jacobian matrix.

A Floquet multiplier is an eigenvalue of a one-period linear evolution map. For a homogeneous linear differential equation with continuous coefficients of period $T>0$, this map is the monodromy matrix of a periodic linear system $N$. If $Nv=mv$ for a nonzero eigenvector $v$, the associated solution of a differential equation with $Y(0)=v$ satisfies $Y(t+T)=mY(t)$. Its exponential rate is $T^{-1}\log |m|$, the corresponding Floquet growth rate. A Floquet multiplier equal to one gives a $T$-periodic solution in that eigenvector direction; a nontrivial Jordan block at a unit-modulus Floquet multiplier can still produce polynomial growth.

For a nonlinear periodic orbit, the Floquet multipliers are eigenvalues of the one-period map of its variational equation. A nonconstant periodic orbit of a smooth autonomous differential equation has a Floquet multiplier equal to one in its tangent direction: differentiating the solution of a differential equation with respect to its starting phase gives a periodic solution of the variational equation. The derivative of a transverse Poincaré map removes this phase direction and retains the transverse Floquet multipliers. Transverse Floquet multipliers strictly inside the unit circle give local attraction, while one outside gives instability; a transverse Floquet multiplier on the unit circle requires further analysis. For a planar periodic orbit, the transverse Floquet multiplier is given by the divergence test for a planar periodic orbit.

A generalized Hopf bifurcation has a vanishing cubic radial coefficient and nonzero quintic coefficient. The two parameters unfold the eigenvalue real part and cubic coefficient. With $R=r^2$, periodic orbits satisfy $\tau +aR+b{R}^2=0$. Their saddle-node bifurcation curve is $\tau =a^2/(4b)$ with $R=-a/(2b)>0$, hence $ab<0$. For $b>0$ the smaller-radius limit cycle is stable and the larger one unstable where both exist; their collision explains why the criticality change of a Hopf bifurcation curve brings a saddle-node bifurcation of periodic orbits.

// Destination: partial-differential-equation.bigb

A global bifurcation changes an invariant set through a global orbit connection or collision that cannot be identified by a single equilibrium point's linearization alone. For example, a global bifurcation through a saddle homoclinic orbit occurs when an outgoing unstable separatrix returns to the same saddle equilibrium's stable manifold. A smooth global Poincaré return map then combines with a local passage map to determine which side contains a periodic orbit. A heteroclinic orbit instead joins distinct equilibrium points; an energy-balance or Melnikov integral can locate its breaking in a nearly Hamiltonian system.

For small positive forcing-frame frequency $\omega$, use $\mu =\hat{\mu }{\omega }^2$, $C=\omega z$ and $\tau =\omega T$ in the three-to-one spatially forced amplitude equation. The leading system is $z_{\tau }+iz={\overline{z}}^2$. Writing $z=x+iy$ gives the Hamiltonian system $(x_{\tau },y_{\tau })=(H_y/2,-H_x/2)$ and the displayed first integral. The origin is a center equilibrium; three saddle equilibria at $(0,1)$ and $(\pm \sqrt{3}/2,-1/2)$ lie on $H=1/3$. The factorization $H-1/3=(1+2y)[x^2-(y-1{)}^2/3]$ reveals a triangular heteroclinic cycle, containing closed periodic orbits for every $0<H<1/3$.

A Hopf bifurcation creates or destroys a small periodic orbit near an equilibrium point as a pair of complex conjugate eigenvalues crosses the imaginary axis. It explains how a resting state can give way to sustained oscillations: the linearization of a dynamical system supplies rotation and growth, while a nonlinear term can saturate that growth at a nonzero amplitude.

The simplest supercritical Hopf bifurcation is the exact Hopf normal formHere $z$ is a complex number, $\mu$ is the bifurcation parameter, and $\omega$ is the angular frequency. In polar coordinates $z=r{e}^{i\theta }$, the equations become $\dot{r}=\mu r-r^3$ and $\dot{\theta }=\omega$ for $r>0$. At $\mu <0$, each turn shrinks towards the equilibrium point. At $\mu >0$, small radii grow and large radii shrink towards $r=\sqrt{\mu }$, producing a limit cycle. For example, $\mu =0.04$ and $\omega =2$ give radius $0.2$, period $\pi$, and frequency $1/\pi$ cycles per unit time. The limiting angular frequency stays nonzero while the amplitude tends to zero: a tiny newborn oscillation can still repeat at a finite rate.

For a precise generic statement, consider a smooth function $f$ defining an autonomous differential equation $\dot{x}=f(x,\alpha )$ on ${\mathbb{R}}^n$, with a smooth branch of equilibrium points $x_{\ast }(\alpha )$ and critical parameter $\alpha =0$. Require:Then a locally unique branch of small periodic orbits, up to phase, emerges on the side $\beta a<0$. On the two-dimensional centre manifold, the branch attracts for $a<0$ and repels for $a>0$. Attraction in the full phase space additionally requires all remaining eigenvalues to have negative real part; any transverse unstable direction makes the periodic orbit unstable there.^[ref]

An eigenvalue-crossing bifurcation test alone cannot establish these nonlinear conclusions. For example, the purely linear equation $\dot{z}=(\mu +i\omega )z$ has the same crossing but no isolated periodic orbit: at $\mu =0$ every nonzero radius is constant, and at $\mu \ne 0$ none is. If the cubic coefficient vanishes, higher-order terms must be examined; a generalized Hopf bifurcation is one possible degeneracy. The usefulness of the generic theorem is that a large system can have its small oscillatory onset understood through one Hopf normal form, subject to these hypotheses and local limits.