# Phase plot of a system of differential equations using Matlab

Assume the system: \begin{align} \begin{pmatrix} x \\ y \\ \end{pmatrix}' &= \begin{pmatrix} 7x+10y+3 \\ -5x-7y+1 \\ \end{pmatrix} \end{align}

By changing the variables so that $$(0,0)$$ is the equilibrium point, the system transforms to: \begin{align} \begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix}' &= \begin{pmatrix} 7x_1+10x_2\\ -5x_1-7x_2 \\ \end{pmatrix} = A \cdot \begin{pmatrix} x_1\\ x_2\\ \end{pmatrix} \end{align} After that, all we have to do in order to determine the equilibrium point's type and stability is to examine $$A$$'s eigenvalues.

Now, I would like to plot this system's solutions and its behavior near the equilibrium point using Matlab. Do I have to use the ode45 command to solve the system and then graph its solutions or is there a quicker way to obtain the phase plot?

Could someone provide me with the code which does that?

You can use quiver for this

[x1, x2] = meshgrid(-1.0 : 0.2 : 1.0, -1.5 : 0.2 : 1.5);
u = 7 * x1 + 10 * x2;
v = -5 * x1 - 7 * x2;

figure
quiver(x1, y1, u, v)


From this representation you can see that solution will circle around $$(x_1, x_2) = (0, 0)$$, clockwise One option is John Polking's pplane.
{"ft_lang_label":"__label__en","ft_lang_prob":0.7946104,"math_prob":0.9995658,"size":847,"snap":"2019-43-2019-47","text_gpt3_token_len":264,"char_repetition_ratio":0.19217081,"word_repetition_ratio":0.0,"special_character_ratio":0.30696577,"punctuation_ratio":0.06410257,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99980575,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-21T08:37:15Z\",\"WARC-Record-ID\":\"<urn:uuid:1eb1c3b8-5c10-4271-b8d9-9cbdfa034875>\",\"Content-Length\":\"139262\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7fd6dc6c-547f-4dd0-86b8-1f7c7d481e44>\",\"WARC-Concurrent-To\":\"<urn:uuid:dd4eccd6-64e9-42f1-a81c-76875cc16b29>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/3017191/phase-plot-of-a-system-of-differential-equations-using-matlab\",\"WARC-Payload-Digest\":\"sha1:S3XGAJNEFXR57RVABTQJPSD2XWA6NSAU\",\"WARC-Block-Digest\":\"sha1:PMMP3E5XVSMPY23PWOKJVPSR7PROUPPT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670743.44_warc_CC-MAIN-20191121074016-20191121102016-00140.warc.gz\"}"}