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We introduce a population-age-time (PAT) model which describes the temporal evolution of the population distribution in age. The surprising result is that the qualitative nature of the population distribution dynamics is robust with respect to the birth rate and death rate distributions in age, and initial conditions. When the number of children born per woman is 2, the population distribution approaches an asymptotically steady state of a kink shape; thus the total population approaches a constant. When the number of children born per woman is greater than 2, the total population increases without bound; and when the number of children born per woman is less than 2, the total population decreases to zero.

We introduced a more general predator-prey model to analyze the paradox of enrichment. We hope the results obtained for the model can guide us on identifying real field paradox of enrichment.

Through a simple and elegant argument, we prove that the norm of the derivative of the solution operator of Euler equations posed in the Sobolev space $H^n$, along any base solution that is in $H^n$ but not in $H^{n+1}$, is infinite. We also review the counterpart of this result for Navier-Stokes equations at high Reynolds number from the perspective of fully developed turbulence. Finally we present a few examples and numerical simulations to show a more complete picture of the so-called rough dependence upon initial data.

Ruelle predicted that the maximal amplification of perturbations in homogeneous isotropic turbulence is exponential $e^{\sigma \sqrt{Re} t}$ (where $\sigma \sqrt{Re}$ is the maximal Liapunov exponent). In our earlier works, we predicted that the maximal amplification of perturbations in fully developed turbulence is faster than exponential $e^{\sigma \sqrt{Re} \sqrt{t} +\sigma_1 t}$. That is, we predicted superfast initial amplification of perturbations. Built upon our earlier numerical verification of our prediction, here we conduct a large numerical verification with resolution up to $2048^3$ and Reynolds number up to $6210$. Our direct numerical simulation here confirms our analytical prediction. Our numerical simulation also demonstrates that such superfast amplification of perturbations leads to superfast nonlinear saturation. We conclude that such superfast amplification and superfast nonlinear saturation of ever existing perturbations serve as the mechanism for the generation, development and persistence of fully developed turbulence.

This article discusses dependence on initial conditions in natural and social sciences with focus on physical science. The main focus is on the newly discovered rough dependence on initial data.

We consider the incompressible 2D Euler equations on bounded spatial domain $S$, and study the solution map on the Sobolev spaces $H^k(S)$ ($k > 2$). Through an elaborate geometric construction, we show that for any $T >0$, the time $T$ solution map $u_0 \mapsto u(T)$ is nowhere locally uniformly continuous and nowhere Fr\'echet differentiable., Comment: 14 pages

We present some new discoveries on the mathematical foundation of linear hydrodynamic stability theory. The new discoveries are: 1. Linearized Euler equations fail to provide a linear approximation on inviscid hydrodynamic stability. 2. Eigenvalue instability predicted by high Reynolds number linearized Navier-Stokes equations cannot capture the dominant instability of super fast growth. 3. As equations for directional differentials, Rayleigh equation and Orr-Sommerfeld equation cannot capture the nature of the full differentials.

Short term unpredictability is discovered numerically for high Reynolds number fluid flows under periodic boundary conditions. Furthermore, the abundance of the short term unpredictability is also discovered. These discoveries support our theory that fully developed turbulence is constantly driven by such short term unpredictability.

We introduce a mathematical model on the dynamics of demand and supply incorporating collectability and saturation factors. Our analysis shows that when the fluctuation of the determinants of demand and supply is strong enough, there is chaos in the demand-supply dynamics. Our numerical simulation shows that such a chaos is not an attractor (i.e. dynamics is not approaching the chaos), instead a periodic attractor (of period 3 under the Poincar\'e period map) exists near the chaos, and co-exists with another periodic attractor (of period 1 under the Poincar\'e period map) near the market equilibrium. Outside the basins of attraction of the two periodic attractors, the dynamics approaches infinity indicating market irrational exuberance or flash crash. The period 3 attractor represents the product's market cycle of growth and recession, while period 1 attractor near the market equilibrium represents the regular fluctuation of the product's market. Thus our model captures more market phenomena besides Marshall's market equilibrium. When the fluctuation of the determinants of demand and supply is strong enough, a three leaf danger zone exists where the basins of attraction of all attractors intertwine and fractal basin boundaries are formed. Small perturbations in the danger zone can lead to very different attractors. That is, small perturbations in the danger zone can cause the market to experience oscillation near market equilibrium, large growth and recession cycle, and irrational exuberance or flash crash.

The mechanism of a rogue water wave is still unknown. One popular conjecture is that the Peregrine wave solution of the nonlinear Schr\"odinger equation (NLS) provides a mechanism. A Peregrine wave solution can be obtained by taking the infinite spatial period limit to the homoclinic solutions. In this article, from the perspective of the phase space structure of these homoclinic orbits in the infinite dimensional phase space where the NLS defines a dynamical system, we exam the observability of these homoclinic orbits (and their approximations). Our conclusion is that these approximate homoclinic orbits are the most observable solutions,and they should correspond to the most common deep ocean waves rather than the rare rogue waves. We also discuss other possibilities for the mechanism of a rogue wave: rough dependence on initial data or finite time blow up.

In this article, we present some explicit solutions showing rough dependence upon initial data. We also studies viscous effects. An extension of a theorem of Cauchy to the viscous case is also presented.

According to a recent theory \cite{Li14}, when the Reynolds number is large, fully developed turbulence is caused by short term unpredictability (rough dependence upon initial data); when the Reynolds number is moderate, often transient turbulence is caused by chaos (long term unpredictability). This article aims at studying chaos in plane Couette flow at moderate Reynolds number. Based upon the work of L. van Veen and G. Kawahara \cite{VK11} on a transversal homoclinic orbit asymptotic to a limit cycle in plane Couette flow, we explore symbolic dynamics and chaos near the homoclinic orbit. Mathematical analysis shows that there is a collection of orbits in the neighborhood of the homoclinic orbit, which is in one-to-one correspondence with the collection of binary sequences. The Bernoulli shift on the binary sequences corresponds to a chaotic dynamics of a properly defined return map.

I propose a new theory on the nature of turbulence: when the Reynolds number is large, violent fully developed turbulence is due to "rough dependence on initial data" rather than chaos which is caused by "sensitive dependence on initial data"; when the Reynolds number is moderate, (often transient) turbulence is due to chaos. The key in the validation of the theory is estimating the temporal growth of the initial perturbations with the Reynolds number as a parameter. Analytically, this amounts to estimating the temporal growth of the norm of the derivative of the solution map of the Navier-Stokes equations, for which here I obtain an upper bound $e^{C \sqrt{t Re} + C_1 t}$. This bound clearly indicates that when the Reynolds number is large, the temporal growth rate can potentially be large in short time, i.e. rough dependence on initial data., Comment: arXiv admin note: text overlap with arXiv:1305.2864

Nowadays, chaos theory and nonlinear dynamics lack research focuses. Here we mention a few major open problems: 1. an effective description of chaos and turbulence, 2. rough dependence on initial data, 3. arrow of time, 4. the paradox of enrichment, 5. the paradox of pesticides, 6. the paradox of plankton.

The paradox of pesticides was observed experimentally, which says that pesticides may dramatically increase the population of a pest when the pest has a natural predator. Here we use a mathematical model to study the paradox. We find that the timing for the application of pesticides is crucial for the resurgence or non-resurgence of the pests. In particular, regularly applying pesticides is not a good idea as also observed in experiments. In fact, the best time to apply pesticides is when the pest population is reasonably high.

A discrete model for computer simulations of the clustering dynamics of Social Amoebae is presented. This model incorporates the wavelike propagation of extracellular signaling cAMP, the sporadic firing of cells at early stage of aggregation, the signal relaying as a response to stimulus, the inertia and purposeful random walk of the cell movement. A Monte Carlo simulation is run which shows the existence of potential equilibriums of mean and variance of aggregation time. The simulation result of this model could well reproduce many phenomena observed by actual experiments.

The paradox of enrichment was observed by M. Rosenzweig in a class of predator-prey models. Two of the parameters in the models are crucial for the paradox. These two parameters are the prey's carrying capacity and prey's half-saturation for predation. Intuitively, increasing the carrying capacity due to enrichment of the prey's environment should lead to a more stable predator-prey system. Analytically, it turns out that increasing the carrying capacity always leads to an unstable predator-prey system that is susceptible to extinction from environmental random perturbations. This is the so-called paradox of enrichment. Our resolution here rests upon a closer investigation on a dimensionless number $H$ formed from the carrying capacity and the prey's half-saturation. By recasting the models into dimensionless forms, the models are in fact governed by a few dimensionless numbers including $H$. The effects of the two parameters: carrying capacity and half-saturation are incorporated into the number $H$. In fact, increasing the carrying capacity is equivalent (i.e. has the same effect on $H$) to decreasing the half-saturation which implies more aggressive predation. Since there is no paradox between more aggressive predation and instability of the predator-prey system, the paradox of enrichment is resolved. We also further explore spatially dependent models for which the phase space is infinite dimensional. The spatially independent limit cycle which is generated by a Hopf bifurcation from an unstable steady state, is linearly stable in the infinite dimensional phase space. Numerical simulations indicate that the basin of attraction of the limit cycle is riddled. This shows that spatial perturbations can sometimes (neither always nor never) remove the paradox of enrichment near the limit cycle!, Comment: submitted

We believe the following three ingredients are enough to explain the mystery of the arrow of time: (1). equations of dynamics of gas molecules, (2). chaotic instabilities of the equations of dynamics, (3). unavoidable perturbations to the gas molecules. The level of physical rigor or mathematical rigor that can be reached for such a theory is unclear., Comment: This note is a continuation of an ongoing project

There are two types of 3D shears in channel flows: ($U(y,z),0,0$) and ($U(y),0,W(y)$). Both are important in organizing the phase space structures of the channel flows. Stability criteria of the type I 3D shears were studied in [Li, 2010]. Here we study the stability criteria of the type II 3D shears. We also provide more support to the idea of resolution of a turbulence paradox, introduced in [Li and Lin, 2010], by studying a sequence of type II 3D shears., Comment: This is a paper very pleasant to write and read

I will prove a recurrence theorem which says that any $H^s$ ($s>2$) solution to the 2D inviscid channel flow returns repeatedly to an arbitrarily small $H^0$ neighborhood. Periodic boundary condition is imposed along the stream-wise direction. The result is an extension of an early result of the author [Li, 09] on 2D Euler equation under periodic boundary conditions along both directions.

We test a Markov chain approximation to the segment description (Li, 2007) of chaos (and turbulence) on a tent map, the Minea system, the H\'enon map, and the Lorenz system. For the tent map, we compute the probability transition matrix of the Markov chain on the segments for segment time length (iterations) $T = 1, 2, 3, 100$. The matrix has $1, 2, 4$ tents corresponding to $T = 1, 2, 3$; and is almost uniform for $T = 100$. As $T \ra +\infty$, our conjecture is that the matrix will approach a uniform matrix (i.e. every entry is the same). For the simple fixed point attractor in the Minea system, the Reynolds average performs excellently and better than the maximal probability Markov chain and segment linking. But for the strange attractors in the H\'enon map, and the Lorenz system, the Reynolds average performs very poorly and worse than the maximal probability Markov chain and segment linking.

Various issues with regard to chaos and recurrence in infinite dimensions are discussed. The doctrine we are trying to derive is that Sobolev spaces over bounded spatial domains do host chaos and recurrence, while Sobolev spaces over unbounded spatial domains are lack of chaos and recurrence. Local Sobolev spaces over unbounded spatial domains can host chaos and are natural phase spaces e.g. for fluid problems, but are very challenging to study., Comment: Relevant manuscripts arXiv:0707.4458 and arXiv:0707.4456 [The Poincar\'e Recurrence Problem of Inviscid Incompressible Fluids, Asian J. Math, vol.13, no.1, 7-14, (2009). A Recurrence Theorem on the Solutions to the 2D Euler Equation Asian J. Math, vol.13, no.1, 1-6, (2009)]

The classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of "streak-roll-critical layer". As the Reynolds number approaches infinity, the steady coherent structure approaches a 3D limiting shear of the form ($U(y,z), 0, 0$) in velocity variables. All such 3D shears are steady states of the 3D Euler equations. This raises the importance of investigating the stability of such inviscid 3D shears in contrast to the classical Rayleigh theory of inviscid 2D shears. Several general criteria of stability for such inviscid 3D shears are derived. In the Appendix, an argument is given to show that a 2D limiting shear can only be the classical laminar shear., Comment: This is a short note. The relevant manuscript is arXiv:0811.0383 (Y. Li, D. Viswanath, Exact and asymptotic conditions on traveling wave solutions of the Navier-Stokes equations, Physics of Fluids 21 (2009), 101703

This is a survey on the recent theory of chaos in partial differential equations.

Flux dynamics in an annular long Josephson junction is studied. Three main topics are covered. The first is chaotic flux dynamics and its prediction via Melnikov integrals. It turns out that DC current bias cannot induce chaotic flux dynamics, while AC current bias can. The existence of a common root to the Melnikov integrals is a necessary condition for the existence of chaotic flux dynamics. The second topic is on the components of the global attractor and the bifurcation in the perturbation parameter measuring the strength of loss, bias and irregularity of the junction. The global attractor can contain co-existing local attractors e.g. a local chaotic attractor and a local regular attractor. In the infinite dimensional phase space setting, the bifurcation is very complicated. Chaotic attractors can appear and disappear in a random fashion. Three types of attractors (chaos, breather, spatially uniform and temporally periodic attractor) are identified. The third topic is ratchet effect. Ratchet effect can be achieved by a current bias field which corresponds to an asymmetric potential, in which case the flux dynamics is ever lasting chaotic. When the current bias field corresponds to a symmetric potential, the flux dynamics is often transiently chaotic, in which case the ratchet effect disappears after sufficiently long time.

Sommerfeld paradox (turbulence paradox) roughly says that mathematically the Couette linear shear flow is linearly stable for all values of the Reynolds number, but experimentally transition from the linear shear to turbulence occurs under perturbations of any size when the Reynolds number is large enough. In [Li, Lin 2011], we offered a resolution of this paradox. The aim of this paper is to provide a numerical implementation of the resolution. The main idea of the resolution is that even though the linear shear is linearly stable, slow orbits (also called quasi-steady states) in arbitrarily small neighborhoods of the linear shear can be linearly unstable. The key is that in infinite dimensions, smallness in one norm does not mean smallness in all norms. Our study here focuses upon a sequence of 2D oscillatory shears which are the Couette linear shear plus small amplitude and high frequency sinusoidal shear perturbations. In the fluid velocity variable, the sequence approaches the Couette linear shear (e.g. in $L^2$ norm of velocity), thus it can be viewed as Couette linear shear plus small noises; while in the fluid vorticity variable, the sequence does not approaches the Couette linear shear (e.g. in $H^1$ norm of velocity). Unlike the Couette linear shear, the sequence of oscillatory shears has inviscid linear instability; furthermore, with the sequence of oscillatory shears as potentials, the Orr-Sommerfeld operator has unstable eigenvalues when the Reynolds number is large enough, this should lead to transient nonlinear growth which manifests as transient turbulence as observed in experiments. The main result of this paper verifies this transient growth., Comment: This is a supporting numerical implementation of the manuscript arXiv:0904.4676

Sommerfeld paradox roughly says that mathematically Couette linear shear is linearly stable for all Reynolds number, but experimentally arbitrarily small perturbations can induce the transition from the linear shear to turbulence when the Reynolds number is large enough. The main idea of our resolution of this paradox is to show that there is a sequence of linearly unstable shears which approaches the linear shear in the kinetic energy norm but not in the enstrophy (vorticity) norm. These oscillatory shears are single Fourier modes in the Fourier series of all the shears. In experiments, such linear instabilities will manifest themselves as transient nonlinear growth leading to the transition from the linear shear to turbulence no matter how small the intitial perturbations to the linear shear are. Under the Euler dynamics, these oscillatory shears are steady, and cat's eye structures bifurcate from them as travelling waves. The 3D shears $U(y,z)$ in a neighborhood of these oscillatory shears are linearly unstable too. Under the Navier-Stokes dynamics, these oscillatory shears are not steady rather drifting slowly. When these oscillatory shears are viewed as frozen, the corresponding Orr-Sommerfeld operator has unstable eigenvalues which approach the corresponding inviscid eigenvalues when the Reynolds number tends to infinity. All the linear instabilities mentioned above offer a resolution to the Sommerfeld paradox, and an initiator for the transition from the linear shear to turbulence., Comment: Based upon colleagues' response, a clarification remark is added at the bottom of Page 3 and the top of Page 4

I shall briefly survey the current status on more rigorous studies of chaos in fluids by focusing along the line of chaos phenotypes: sensitive dependence on initial data, and recurrence.